Calculation method, generation method, program, exposure method, and mask fabrication method

ABSTRACT

A generation method of generating, by a computer, data of a pattern of a mask used for an exposure apparatus including a projection optical system. The method includes dividing an effective light source formed on a pupil plane of the projection optical system into a plurality of point sources; generating a plurality of shifted pupil functions by shifting a pupil function corresponding to each of the plurality of point sources by a shift amount in accordance with a position of each point source; defining a matrix by arranging each of the plurality of shifted pupil functions in each row or each column of the matrix; calculating an eigenvalue and an eigenfunction by performing singular value decomposition of the matrix; calculating a map representing, when elements of a target pattern are inserted on an object plane of the projection optical system, an influence the elements inflict on each other.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a Divisional application pursuant to 37 CFR §1.53(b)of copending U.S. patent application Ser. No. 13/248,480 filed Sep. 29,2011, which is a divisional of prior U.S. application Ser. No.12/241,702 filed Sep. 30, 2008, now U.S. Pat. No. 8,059,262, whichclaims foreign priority benefit from Japanese Patent Application No.2007-260360 filed on Oct. 3, 2007. The disclosures of the above-namedapplications are hereby incorporated by reference herein in theirentirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a calculation method, a generationmethod, a program, an exposure method, and a mask fabrication method.

2. Description of the Related Art

A projection exposure apparatus which projects and transfers a circuitpattern formed on a mask (or reticle) onto a substrate such as a waferby a projection optical system is employed to fabricate a semiconductordevice by using photolithography. Along with the recent advance in themicropatterning of semiconductor devices, the projection exposureapparatus is being desired to further improve the resolving power(attain a higher resolution).

As a means for achieving a higher resolution of the projection exposureapparatus, it is a common practice to attain a higher NA of theprojection optical system (increasing the numerical aperture (NA) of theprojection optical system), and to shorten the exposure light. Also, theRET (Resolution Enhanced Technology) which improves the resolution ofthe projection exposure apparatus by decreasing the k1 factor (alsocalled the “process constant”) is attracting a great deal of attention.

The smaller the k1 factor, the higher the degree of difficulty ofexposure. Conventionally, exposure conditions under which a circuitpattern can be projected faithfully have been detected by repeatingexperiments several times. That is, exposure (e.g., the exposureconditions and exposure method) has been optimized in this way. Atpresent, however, as the degree of difficulty of exposure is increasing,the detection of the exposure conditions based on experiments requires along time and high cost. Nowadays, to solve this problem, it is becomingmainstream to optimize, for example, the exposure conditions byrepeating exposure simulation using a computer. The mainstream of thesimulation technique is the so-called model-based RET which executessimulation based on a physical model of optics.

The model-based RET generally uses partial coherent imaging calculation.Improving the speed of the partial coherent imaging calculation makes itpossible to shorten the time taken for the model-based RET. Nowadays,along with the progress in computer environment, the calculation speedis improved by forming a parallel processing system using a plurality ofcomputers. There has also been proposed a technique of improving thecalculation speed more effectively than in the formation of a parallelprocessing system using computers by improving an algorithm whichexecutes the partial coherent imaging calculation.

For example, Cris Spence, “Full-Chip Lithography Simulation and DesignAnalysis—How OPC is Changing IC Design”, Proceedings of SPIE, U.S.A.,SPIE press, 2005, Vol. 5751, pp. 1-14 reports that an algorithm calledthe SOCS increased the calculation speed (simulation speed) to 10,000times that before. Also, Alfred Kwok-kit Wong, “Optical Imaging inProjection Microlithography”, U.S.A., SPIE press, 2005, pp. 151-163describes the partial coherent imaging calculation, but does notintroduce an algorithm which attains a calculation speed more than thatattained by using the SOCS algorithm. Note that Alfred Kwok-kit Wong,“Optical Imaging in Projection Microlithography”, U.S.A., SPIE press,2005, pp. 151-163 calls the SOCS coherent decomposition.

Unfortunately, the SOCS requires much time to calculate the TCC(Transmission Cross Coefficient) and decompose it into eigenvalues andeigenfunctions.

SUMMARY OF THE INVENTION

The present invention provides a calculation method which can calculatethe TCC in an exposure apparatus in a short period of time. The presentinvention provides a calculation method which can calculate a lightintensity distribution, which is formed on the image plane of aprojection optical system, in a short period of time. The presentinvention provides a generation method which can generate data of apattern of a mask in a short period of time.

According to the first aspect of the present invention, there isprovided a calculation method of calculating, by a computer, a lightintensity distribution formed on an image plane of a projection opticalsystem upon illuminating a mask using an illumination optical system andprojecting an image of a pattern of the mask onto a substrate via theprojection optical system, comprising a division step of dividing aneffective light source formed on a pupil plane of the projection opticalsystem into a plurality of point sources, a generation step of shiftinga pupil function describing a pupil of the projection optical system foreach of the plurality of point sources in accordance with positionsthereof, thereby generating a plurality of shifted pupil functions, adefining step of defining a matrix including the plurality of pupilfunctions generated in the generation step, a first calculation step ofperforming singular value decomposition of the matrix defined in thedefining step, thereby calculating an eigenvalue and an eigenfunction,and a second calculation step of calculating the light intensitydistribution formed on the image plane of the projection optical system,based on a distribution of the light diffracted by the pattern of themask, and the eigenvalue and the eigenfunction calculated in the firstcalculation step.

According to the second aspect of the present invention, there isprovided a generation method of generating, by a computer, data of apattern of a mask used for an exposure apparatus including a projectionoptical system, comprising a division step of dividing an effectivelight source formed on a pupil plane of the projection optical systeminto a plurality of point sources, a generation step of shifting a pupilfunction describing a pupil of the projection optical system for each ofthe plurality of point sources in accordance with positions thereof,thereby generating a plurality of shifted pupil functions, a definingstep of defining a matrix including the plurality of pupil functionsgenerated in the generation step, a first calculation step of performingsingular value decomposition of the matrix defined in the defining step,thereby calculating an eigenvalue and an eigenfunction, a secondcalculation step of calculating a map representing, when elements of atarget pattern are inserted on an object plane of the projection opticalsystem, an influence the elements inflict on each other, based on adistribution of the light diffracted by the target pattern, and theeigenvalue and the eigenfunction calculated in the first calculationstep, and a data generation step of generating data of the pattern ofthe mask based on the map calculated in the second calculation step.

According to the third aspect of the present invention, there isprovided a storage medium storing a program for making a computerexecute a process of calculating a light intensity distribution formedon an image plane of a projection optical system upon illuminating amask using an illumination optical system and projecting an image of apattern of the mask onto a substrate via the projection optical system,the program making the computer execute a division step of dividing aneffective light source formed on a pupil plane of the projection opticalsystem into a plurality of point sources, a generation step of shiftinga pupil function describing a pupil of the projection optical system foreach of the plurality of point sources in accordance with positionsthereof, thereby generating a plurality of shifted pupil functions, adefining step of defining a matrix including the plurality of pupilfunctions generated in the generation step, a first calculation step ofperforming singular value decomposition of the matrix defined in thedefining step, thereby calculating an eigenvalue and an eigenfunction,and a second calculation step of calculating the light intensitydistribution formed on the image plane of the projection optical system,based on a distribution of the light diffracted by the pattern of themask, and the eigenvalue and the eigenfunction calculated in the firstcalculation step.

According to the fourth aspect of the present invention, there isprovided a storage medium storing a program for making a computerexecute a process of generating data of a pattern of a mask used for anexposure apparatus including a projection optical system, the programmaking the computer execute a division step of dividing an effectivelight source formed on a pupil plane of the projection optical systeminto a plurality of point sources, a generation step of shifting a pupilfunction describing a pupil of the projection optical system for each ofthe plurality of point sources in accordance with positions thereof,thereby generating a plurality of shifted pupil functions, a definingstep of defining a matrix including the plurality of pupil functionsgenerated in the generation step, a first calculation step of performingsingular value decomposition of the matrix defined in the defining step,thereby calculating an eigenvalue and an eigenfunction, a secondcalculation step of calculating a map representing, when elements of atarget pattern are inserted on an object plane of the projection opticalsystem, an influence the elements inflict on each other, based on adistribution of the light diffracted by the target pattern, and theeigenvalue and the eigenfunction calculated in the first calculationstep, and a data generation step of generating data of the pattern ofthe mask based the map calculated in the second calculation step.

According to the fifth aspect of the present invention, there isprovided an exposure method comprising a calculation step of calculatinga light intensity distribution formed on an image plane of a projectionoptical system upon illuminating a mask using an illumination opticalsystem and projecting an image of a pattern of the mask onto a substratevia the projection optical system, an adjusting step of adjusting anexposure condition based on the light intensity distribution calculatedin the calculation step, and an exposure step of projecting the image ofthe pattern of the mask onto the substrate after the adjusting step, thecalculation step including a division step of dividing an effectivelight source formed on a pupil plane of the projection optical systeminto a plurality of point sources, a generation step of shifting a pupilfunction describing a pupil of the projection optical system for each ofthe plurality of point sources in accordance with positions thereof,thereby generating a plurality of shifted pupil functions, a definingstep of defining a matrix including the plurality of pupil functionsgenerated in the generation step, a first calculation step of performingsingular value decomposition of the matrix defined in the defining step,thereby calculating an eigenvalue and an eigenfunction, and a secondcalculation step of calculating the light intensity distribution formedon the image plane of the projection optical system, based on adistribution of the light diffracted by the pattern of the mask, and theeigenvalue and the eigenfunction calculated in the first calculationstep.

According to the sixth aspect of the present invention, there isprovided a mask fabrication method comprising generating data of apattern for a mask by the above generation method, and fabricating themask using generated data.

According to the seventh aspect of the present invention, there isprovided an exposure method comprising steps of fabricating a mask bythe above mask fabrication method, illuminating the fabricated mask, andprojecting an image of a pattern of the mask onto a substrate via aprojection optical system.

According to the eighth aspect of the present invention, there isprovided a calculation method of calculating, by a computer, atransmission cross coefficient in an exposure apparatus whichilluminates a mask using an illumination optical system and projects animage of a pattern of the mask onto a substrate via a projection opticalsystem, comprising a division step of dividing an effective light sourceformed on a pupil plane of the projection optical system into aplurality of point sources, a generation step of shifting a pupilfunction describing a pupil of the projection optical system for each ofthe plurality of point sources in accordance with positions thereof,thereby generating a plurality of shifted pupil functions, a definingstep of defining a matrix including the plurality of pupil functionsgenerated in the generation step, and a calculation step of calculatingthe transmission cross coefficient based on the matrix defined in thedefining step.

Further features of the present invention will become apparent from thefollowing description of exemplary embodiments with reference to theattached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic block diagram showing the configuration of aprocessing apparatus 1 which executes a calculation method according toone aspect of the present invention.

FIG. 2 is a chart schematically showing one-dimensional plane waves(orthogonal function system).

FIG. 3 is a flowchart for explaining details of a process of calculatingan aerial image by an aerial image calculation program in the processingapparatus 1 shown in FIG. 1.

FIGS. 4A to 4D are charts for explaining the first embodiment accordingto the present invention, in which FIG. 4A shows an effective lightsource used in the first embodiment, FIG. 4B shows mask data used in thefirst embodiment, FIG. 4C shows an aerial image calculated by an aerialimage calculation program, and FIG. 4D shows an aerial image calculatedby the SOCS.

FIG. 5 is a chart for explaining the second embodiment according to thepresent invention, which shows an aerial image calculated by an aerialimage calculation program when a projection optical system hasaberration.

FIG. 6 is a chart for explaining the second embodiment according to thepresent invention, which shows an aerial image calculated by the aerialimage calculation program when illumination light is polarized.

FIGS. 7A to 7C are charts for explaining the third embodiment accordingto the present invention, in which FIG. 7A shows mask data and FIGS. 7Band 7C show aerial images calculated by an aerial image calculationprogram using the mask data shown in FIG. 7A and that after the OPC.

FIGS. 8A and 8B are graphs for explaining the fourth embodimentaccording to the present invention, in which FIG. 8A shows therelationship between the number and square of the eigenvalue, and FIG.8B shows the difference between a complete aerial image and anapproximated aerial image (i.e., an aerial image calculated from someeigenvalues and eigenfunctions).

FIGS. 9A to 9C are charts for explaining the fourth embodiment accordingto the present invention, in which FIG. 9A shows an effective lightsource, FIG. 9B shows an uncompressed P operator, and FIG. 9C shows acompressed P operator.

FIG. 10 is a block diagram showing the configuration of a processingapparatus according to the fifth embodiment of the present invention.

FIGS. 11A to 11E are charts for explaining the fifth embodimentaccording to the present invention, in which FIG. 11A shows an effectivelight source, FIG. 11B shows pattern data, FIG. 11C shows a P map, FIG.11D shows mask data, and FIG. 11E shows regions each of which exhibits avalue equal to or more than a threshold on the P map.

FIG. 12 is a graph showing the result of a comparison of the imagingperformances of a mask having no assist patterns, that in which assistpatterns are inserted according to the prior art, and that in whichassist patterns are inserted according to the fifth embodiment.

FIGS. 13A and 13B are charts for explaining the sixth embodimentaccording to the present invention, in which FIG. 13A shows pattern dataand FIG. 13B shows a P map.

FIGS. 14A to 14C are charts for explaining the seventh embodimentaccording to the present invention, in which FIG. 14A shows an effectivelight source, FIG. 14B shows a P map, and FIG. 14C shows a mask.

FIGS. 15A to 15F are charts for explaining the eighth embodimentaccording to the present invention, which show P maps.

FIGS. 16A and 16B are charts for explaining the ninth embodimentaccording to the present invention, in which FIG. 16A shows a P map andFIG. 16B shows mask data.

FIG. 17 is a graph showing the result of a comparison of the imagingperformances of a mask based on the mask data shown in FIG. 11D (fifthembodiment), and that based on the mask data shown in FIG. 16B (ninthembodiment).

FIG. 18 is a chart showing mask data according to the tenth embodimentof the present invention.

FIG. 19 is a graph showing the result of a comparison of the imagingperformances of a mask based on the mask data shown in FIG. 16B (ninthembodiment), and that based on the mask data shown in FIG. 18 (tenthembodiment).

FIG. 20 is a graph for explaining the eleventh embodiment according tothe present invention, which shows the defocus characteristic whenassist patterns that are in phase with a desired pattern to betransferred by exposure are arranged at positions having positive valueson a P map, and that when assist patterns that are in phase with thedesired pattern are arranged at positions having negative values on theP map.

FIG. 21 is a flowchart for explaining a process of generating patterndata free from any forbidden pitch by a mask generation program.

FIGS. 22A to 22E are charts for explaining the eleventh embodimentaccording to the present invention, in which FIG. 22A shows patterndata, FIGS. 22B and 22C show mask data, and FIGS. 22D and 22E show maskdata in which assist patterns are inserted.

FIGS. 23A to 23D are charts for explaining the twelfth embodimentaccording to the present invention, in which FIG. 23A shows patterndata, FIG. 23B shows an effective light source, FIG. 23C shows a P map,and FIG. 23D shows mask data.

FIG. 24 is a flowchart for explaining a mask data generation process bya mask generation program.

FIGS. 25A and 25B are charts for explaining the twelfth embodimentaccording to the present invention, in which FIG. 25A shows pattern dataand FIG. 25B shows an effective light source.

FIG. 26 is a schematic block diagram showing the configuration of anexposure apparatus according to one aspect of the present invention.

DESCRIPTION OF THE EMBODIMENTS

Preferred embodiments of the present invention will be described belowwith reference to the accompanying drawings. The same reference numeralsdenote the same members throughout the drawings, and a repetitivedescription thereof will not be given.

For example, the present invention is applicable to optical systemimaging calculation based on partial coherent imaging (partial coherentimaging calculation) in, for example, an exposure apparatus andmicroscope. The present invention is also applicable to the generationof data of a mask used in micromechanics and in fabricating variousdevices, for example, semiconductor chips such as an IC and LSI, adisplay device such as a liquid crystal panel, a detection device suchas a magnetic head, and an image sensing device such as a CCD. Themicromechanics means herein a technique of fabricating amicrometer-order sophisticated machinery system by applying asemiconductor integrated circuit fabrication technique to thefabrication of a microstructure, or the machinery system itself.

The concept disclosed in the present invention can be modeledmathematically. Hence, the present invention can be implemented as asoftware function of a computer system. A software function of acomputer system includes programming having executable software codes,and executes partial coherent imaging calculation in this embodiment.The software codes are executed by the processor of the computer system.Codes or associated data records are stored in the computer platformduring the software code operation. However, the software codes areoften stored in other sites or loaded into an appropriate computersystem. The software codes can be held on at least one computer-readablerecording medium as one or a plurality of modules. The contents of thepresent invention can be described in the form of codes described above,which can function as one or a plurality of software products.

Coordinate systems in an exposure apparatus according to this embodimentwill be explained first. In this embodiment, the coordinate systems inan exposure apparatus are roughly classified into two.

The first coordinate system defines the coordinates on the mask surface(the object plane of the projection optical system) and the wafersurface (the image plane of the projection optical system), which areexpressed by (x, y) in this embodiment. The pattern size on the masksurface is different from that on the wafer surface by the magnificationof the projection optical system. For the sake of descriptivesimplicity, the ratio between the pattern size on the mask surface andthat on the wafer surface is set at 1:1 by multiplying the pattern sizeon the mask surface by the magnification of the projection opticalsystem in the following description. With this setting, the ratiobetween the coordinate system on the mask surface and that on the wafersurface also becomes 1:1.

The second coordinate system defines the coordinates on the pupil planeof the projection optical system, which are expressed by (f, g) in thisembodiment. The coordinates (f, g) on the pupil plane of the projectionoptical system are defined by a coordinate system normalized assumingthat the pupil size of the projection optical system is 1.

In the exposure apparatus, a light intensity distribution formed on thepupil plane of the projection optical system while no mask is insertedon the object plane of the projection optical system is called aneffective light source, which is expressed by S(f, g) in thisembodiment. The pupil of the projection optical system is expressed by apupil function P(f, g) in this embodiment. In general, the pupilfunction can include the influences (pieces of information) ofaberration and polarization on the pupil property. Even the pupilfunction P(f, g) in this embodiment can include the influences ofaberration and polarization on the pupil property.

The exposure apparatus illuminates a mask by partial coherentillumination and projects the pattern of the mask (mask pattern) onto awafer. In this embodiment, a mask pattern including pieces ofinformation on the transmittance and phase is defined by o(x, y), and alight intensity distribution (aerial image) formed on the wafer surfaceis defined by I(x, y). The amplitude of light diffracted by the maskpattern is defined by the pupil plane of the projection optical system,and expressed by a(f, g) in this embodiment.

The conventional partial coherent imaging calculation will be explainedherein. The conventional partial coherent imaging calculation (thecalculation of the light intensity distribution on the image plane ofthe projection optical system) can be roughly classified into threetypes.

The first calculation method is the so-called Abbe method. Morespecifically, the Abbe method calculates the light intensitydistribution I(x, y) by:

$\begin{matrix}{{I( {x,y} )} = {\sum\limits_{i = 1}^{N_{1}}\; {{S( {f_{i}^{\prime},g_{i}^{\prime}} )}{{F\lbrack {{P( {f,g} )}{a( {{f - f_{i}^{\prime}},{g - g^{\prime}}} )}} \rbrack}}^{2}}}} & (1)\end{matrix}$

where N₁ is the number of point sources for numerical calculation, and Fis the Fourier transform.

The second calculation method calculates the TCC without eigenvaluedecomposition. The TCC is defined by:

TCC(f′,g′,f″,g″)=∫∫S(f,g)P(f+f′,g+g′)P*(f+f″,g+g″)dfdg  (2)

Referring to equation (2), the TCC is given by a four-dimensionalfunction. Using the TCC, the light intensity distribution I(x, y) can becalculated by:

$\begin{matrix}{{I( {x,y} )} = {\sum\limits_{i,j,k,{l = 1}}^{N_{2}}\; {{{TCC}( {f_{i}^{\prime},g_{j}^{\prime},f_{k}^{''},g_{l}^{''}} )}{a( {f_{i}^{\prime},g_{j}^{\prime}} )}{a^{*}( {f_{k}^{''},g_{l}^{''}} )} \times \exp \{ {{- }\; 2\; {\pi \lbrack {{( {f_{i}^{\prime} - f_{k}^{''}} )x} + {( {g_{j}^{\prime} - g_{l}^{''}} )y}} \rbrack}} \}}}} & (3)\end{matrix}$

where N₂ is a number that i, j, k, and l can take, and depends on thepupil division number for numerical calculation.

The third calculation method is the so-called SOCS, which decomposes theTCC expressed by equation (2) into a plurality of eigenvalues andeigenfunctions. The light intensity distribution I(x, y) is calculatedby:

$\begin{matrix}{{I( {x,y} )} = {\sum\limits_{i = 1}^{N_{3}}\; {\lambda_{i}{{F\lbrack {{\psi_{i}( {f,g} )}{a( {f,g} )}} \rbrack}}^{2}}}} & (4)\end{matrix}$

where λ_(i) is the i-th eigenvalue, ψ_(i) is the i-th eigenfunction, andN₃ is the number of point sources for numerical calculation.

The Abbe method is suitable for small-scale calculation (small-scalesimulation). More specifically, the Abbe method is suitable forsimulation associated with part of a mask, and for checking changes inimaging performance when optical settings (e.g., the effective lightsource, aberration, and polarization) are changed.

The calculation speed in the calculation method using the TCC, that is,the calculation method using equation (3) is lower than those in theAbbe method and SOCS because quadrupole integration must be performed inequation (3). To calculate the light intensity distribution without thequadrupole integration in equation (3), the SOCS is available. The SOCSis suitable for large-scale calculation (large-scale simulation).

In large-scale calculation, the partial coherent imaging calculation isperformed by dividing a mask into a plurality of regions. If opticalsettings do not change, the TCC expressed by equation (2) does notchange and the eigenfunction ψ_(i) in equation (4), in turn, does notchange either. Once the eigenvalue λ_(i) and eigenfunction ψ_(i) arecalculated, simple calculation need only be repeated thereafter, so theSOCS is suitable for large-scale calculation. However, the SOCS isunsuitable for small-scale calculation.

As can be understood from equation (2), since double integration isnecessary to calculate the TCC (i.e., the TCC is given by afour-dimensional function), the SOCS requires much time to calculate theTCC and a huge capacity of computer memory. The SOCS also requires muchtime to calculate the eigenvalue λ_(i) and eigenfunction ψ_(i).Furthermore, if optical settings change, the TCC must be calculatedagain in the SOCS. From these viewpoints, the SOCS is unsuitable forchecking changes in imaging performance by changing the opticalsettings.

As described above, the conventional calculation methods require hugeamounts of time for simulation. In addition, the Abbe method and SOCSmust be selectively used in accordance with the calculation target(i.e., whether small-scale calculation or large-scale calculation isexecuted) in the prior art.

FIG. 1 is a schematic block diagram showing the configuration of aprocessing apparatus 1 which executes a calculation method according toone aspect of the present invention.

The processing apparatus 1 is formed from, for example, ageneral-purpose computer and includes a bus line 10, control unit 20,display unit 30, storage unit 40, input unit 50, and medium interface60, as shown in FIG. 1.

The bus line 10 interconnects the control unit 20, display unit 30,storage unit 40, input unit 50, and medium interface 60.

The control unit 20 is formed from a CPU, GPU, DSP, or microcomputer,and includes a cash memory for temporal storage.

The display unit 30 is formed from, for example, a display device suchas a CRT display or liquid crystal display.

The storage unit 40 is formed from, for example, a memory or hard disk.In this embodiment, the storage unit 40 stores pattern data 401,effective light source information 402, NA information 403, Ainformation 404, aberration information 405, polarization information406, and resist information 407. The storage unit 40 also stores a Poperator 408, an aerial image 409, mask data 410, and an aerial imagecalculation program 411.

The pattern data 401 is data of a pattern (layout pattern or targetpattern) laid out in designing, for example, an integrated circuit.

The effective light source information 402 is associated with a lightintensity distribution (effective light source) formed on the pupilplane of the projection optical system of the exposure apparatus.

The NA information 403 is associated with the numerical aperture, on theimage side, of the projection optical system of the exposure apparatus.

The λ information 404 is associated with the wavelength of light(exposure light) emitted by the light source of the exposure apparatus.

The aberration information 405 is associated with the aberration of theprojection optical system of the exposure apparatus.

The polarization information 406 is associated with the polarization oflight formed by the illumination apparatus (illumination optical system)of the exposure apparatus (the polarization state of the illuminationlight).

The resist information 407 is associated with a resist applied on thewafer.

The P operator 408 is a matrix necessary in the process of calculatingan aerial image as a light intensity distribution formed on the wafersurface (i.e., for the aerial image calculation program 411), as will bedescribed in detail later.

The aerial image 409 is the result of calculating the aerial image(light intensity distribution) by the aerial image calculation program411.

The mask data 410 is data of an actual mask (reticle). The mask data 410is generally different from the pattern data 401.

The aerial image calculation program 411 is a program for calculatingthe aerial image (light intensity distribution).

The input unit 50 includes, for example, a keyboard and mouse.

The medium interface 60 includes, for example, a floppy disk drive,CD-ROM drive, and USB interface, and can be connected to a storagemedium 70. The storage medium 70 includes, for example, a floppy disk,CD-ROM, and USB memory.

How to calculate the aerial image 409 by the aerial image calculationprogram 411 will be explained below by paying attention particularly tothe P operator 408. Note that in this embodiment, the wavelength of theexposure light is indicated by A, and the numerical aperture, on theimage side, of the projection optical system is indicated by NA. Notealso that the ratio between the numerical aperture of illumination lightwhich is guided from the illumination optical system to the mask surfaceand that of the projection optical system on its object side isindicated by σ.

The mask pattern and the aerial image in the exposure apparatus have apartial coherent imaging relationship. The partial coherent imagingcalculation is roughly classified into three types (see equations (1),(3), and (4)), as described above. Since the Fourier transforms F areused in equations (1) and (4), the sum of plane waves forms an aerialimage from the viewpoint of Fourier optics. Each plane wave is expressedby exp[−i2π(fx+gy)]. Although equation (3) does not clearly exhibit theFourier transform F, the sum of plane waves similarly forms an aerialimage because exp[−i2π(fx+gy)] is included in it.

In this manner, the partial coherent imaging is based on the plane waveexp[−i2π(fx+gy)] from the viewpoint of optics. On the other hand,exp[−i2π(fx+gy)] is defined by an orthogonal function system from theviewpoint of mathematics. In this embodiment, the plane wave is definedby an orthogonal function system, thereby attaining the calculation ofthe aerial image 409 in a shorter period of time.

A case in which a one-dimensional aerial image (light intensitydistribution) is calculated will be exemplified first. In this case, theplane wave can be expressed by exp(−i2πfx). The orthogonal functionsystem is defined by a vector:

$\begin{matrix}{{\varphi\rangle} = \begin{pmatrix}^{{- {2\pi}}\; f_{1}x} \\^{{- {2\pi}}\; f_{2}x} \\\vdots \\^{{- {2\pi}}\; f_{M}x}\end{pmatrix}} & (5)\end{matrix}$

where M is the division number of f when −2≦f≦2.

The P operator 408 will be explained herein. Since M in equation (5) isassumed as 7 in this embodiment, f₁=−2, f₂=−4/3, f₃=−2/3, f₄=0, f₅=2/3,f₆=4/3, and f₇=2, as shown in FIG. 2. FIG. 2 is a chart schematicallyshowing one-dimensional plane waves (orthogonal function system).

The distribution of light diffracted by the mask pattern (diffractedlight distribution) can be expressed by a(f_(i))exp(−i2πf_(i)x). Then, avector |φ′> of the diffracted light distribution can be expressed by:

$\begin{matrix}\begin{matrix}{{\varphi^{\prime}\rangle} = {\begin{pmatrix}{a( f_{1} )} & 0 & \ldots & 0 \\0 & {a( f_{2} )} & \; & 0 \\\vdots & \; & \ddots & 0 \\0 & 0 & \ldots & {a( f_{7} )}\end{pmatrix}\begin{pmatrix}^{{- 2}\pi \; f_{1}x} \\^{{- 2}\pi \; f_{2}x} \\\vdots \\^{{- 2}\pi \; f_{7}x}\end{pmatrix}}} \\{= {A{\varphi\rangle}}}\end{matrix} & (6)\end{matrix}$

where A is a diagonal matrix having the amplitude a(f_(i)) of thediffracted light as the diagonal element.

When the projection optical system has no aberration, its pupil has afunction of passing a diffracted light component in the range of −1≦f≦1intact, and shielding that in the range of |f|>1. Outputting light fromone point f′ on the effective light source amounts to shifting the pupilof the projection optical system by f′. Hence, when light emanating fromone point f′ on the effective light source is diffracted by the maskpattern, a diffracted light component in the range of −1≦f−f′≦1 passesthrough the pupil of the projection optical system, while that in therange of |f−f′|>1 is shielded by the pupil of the projection opticalsystem.

For example, if light emanating from f=f₄=0 on the effective lightsource is diffracted by the mask pattern and stopped down by the pupilof the projection optical system, an amplitude |φ₁> of the diffractedlight transmitted through the pupil of the projection optical system canbe expressed by:

|φ

=(0011100)A|φ

  (7)

Calculating the square of the absolute value of the amplitude of thediffracted light transmitted through the pupil of the projection opticalsystem yields the light intensity on the wafer surface. Hence, a lightintensity distribution I₁(x) formed on the wafer surface by the pointsource at f=f₄=0 can be expressed by:

I ₁(x)=

φ₁|φ₁

  (8)

where <φ₁| is the transposed conjugate (adjoint) matrix of |φ₁>.

Likewise, if light emanating from f=f₃ on the effective light source isdiffracted by the mask pattern and stopped down by the pupil of theprojection optical system, an amplitude |φ₂> of the diffracted lighttransmitted through the pupil of the projection optical system can beexpressed by:

|φ₂

=(0001110)A|φ

  (9)

Hence, a light intensity distribution I₂(x) formed on the wafer surfaceby the point source at f=f₃ can be expressed by:

I ₂(x)=

φ₂|φ₂

  (10)

Also, the partial coherent illumination can be considered to be a set ofincoherent point sources. For example, assume that two point sourcesexist on the effective light source, and the coordinates of these pointsources are f=0 and f=f₃. Since the two point sources are incoherent, alight intensity distribution I(x) formed on the wafer surface by thesetwo point sources can be expressed by I₁(x)+I₂(x) (i.e., the sum of thelight intensities on the wafer surface).

A P operator P_(1D) is defined by:

$\begin{matrix}{P_{1D} = \begin{pmatrix}0011100 \\0001110\end{pmatrix}} & (11)\end{matrix}$

Referring to equation (11), each row of the P operator P_(1D) is avector in which the pupil of the projection optical system is shifted inaccordance with the position of each point source on the effective lightsource. More specifically, the pupil of the projection optical systemneed only be shifted by the difference between the central position onthe pupil plane of the projection optical system and the position ofeach point source. Using the P operator P_(1D), a light intensitydistribution I(x) formed on the wafer surface can be expressed by:

I(x)=

φ′|P _(1D) ⁺ P _(1D)|φ′

  (12)

Note that the “+” sign represents the transposed conjugate matrix of acertain matrix. Referring to equation (12), the light intensitydistribution I(x) is I₁(x)+I₂(x). In other words, the use of the Poperator P_(1D) allows to simply express an aerial image as a lightintensity distribution formed on the wafer surface.

Equation (12) can be rewritten as:

I(x)=

φ′|T _(1D)|φ′

  (13)

T_(1D) is a matrix defined by:

T _(1D) =P _(1D) ⁺ P _(1D)  (14)

The matrix T_(1D) defined by equation (14) describes the TCC. Tocalculate P_(1D), the pupil of the projection optical system need onlybe shifted and multiplication and addition are not needed. This makes itpossible to calculate P_(1D) in a shorter period of time. Still better,since the TCC can be calculated by the multiplication of P_(1D) and itstransposed conjugate, the use of the P operator 408 allows to calculatethe TCC more quickly than in the use of equation (2).

Note that P_(1D) is not a square matrix. Using the singular valuedecomposition, P_(1D) is rewritten as:

P _(1D) =WSV  (15)

where S is a diagonal matrix, and W and V are unitary matrices.Substituting equation (15) into equation (12), and solving it using thetheorem of the singular value decomposition that W⁺W is a unit matrixyields:

$\begin{matrix}\begin{matrix}{{I(x)} = {\langle{\varphi^{\prime}{{P_{1D}^{+}P_{1D}}}\varphi^{\prime}}\rangle}} \\{= {\langle{\varphi^{\prime}{{V^{+}S^{+}W^{+}{WSV}}}\varphi^{\prime}}\rangle}} \\{= {\langle{\varphi^{\prime}{{V^{+}{SSV}}}\varphi^{\prime}}\rangle}} \\{= {S^{2}{\langle{\varphi^{\prime}{{V^{+}V}}\varphi^{\prime}}\rangle}}} \\{= {S^{2}{\langle{\Phi \Phi}\rangle}}}\end{matrix} & (16)\end{matrix}$

The SOCS as one conventional partial coherent imaging calculation methoddecomposes the TCC into eigenvalues and eigenfunctions, as describedabove. Since the TCC is a very large matrix, a huge amount of time and ahuge capacity of computer memory are required to calculate the TCC.Furthermore, a huge amount of time is also required to decompose the TCCinto eigenvalues and eigenfunctions.

In this embodiment, the singular value decomposition is performed forthe P operator 408. Referring to equation (14), since the elements ofthe P operator 408 are obviously fewer than those of the TCC, thesingular value decomposition of the P operator 408 requires less timethan in the TCC. Moreover, since multiplication and addition are notneeded to calculate the P operator 408, it is possible to calculate theP operator 408 in a shorter period of time. In other words, the use ofthe P operator 408 allows to calculate eigenvalues and eigenfunctionswith a smaller amount of calculation and a smaller memory capacity thanin the SOCS. This makes it possible to calculate the aerial image 409 asa light intensity distribution, which is formed on the wafer surface, ina shorter period of time. Also, the use of equation (14) allowscalculating the TCC in a shorter period of time.

A case in which a one-dimensional aerial image (light intensitydistribution) is calculated has been exemplified above, and a case inwhich a two-dimensional aerial image (light intensity distribution) iscalculated will be exemplified below.

Let (f_(i), g_(j)) be the coordinates on the pupil plane of thediscretized projection optical system. Note that i and j range from 1 toM. A vector |φ′_(2D)> of the diffracted light distribution is expressedby a one-dimensional array of elements:

$\begin{matrix}{{\varphi_{2D}^{\prime}\rangle} = \begin{pmatrix}{{a( {f_{1},g_{1}} )}^{- {{2\pi}{({{f_{1}x} + {g_{1}y}})}}}} \\{{a( {f_{2},g_{1}} )}^{- {{2\pi}{({{f_{2}x} + {g_{1}y}})}}}} \\\vdots \\{{a( {f_{m},g_{1}} )}^{- {{2\pi}{({{f_{M}x} + {g_{1}y}})}}}} \\{{a( {f_{1},g_{2}} )}^{- {{2\pi}{({{f_{1}x} + {g_{2}y}})}}}} \\\vdots \\{{a( {f_{M},g_{M}} )}^{- {{2\pi}{({{f_{M}x} + g_{M\; y}})}}}}\end{pmatrix}} & (17)\end{matrix}$

For the sake of a detailed expression of |φ′_(2D)>, “floor” representsthe omission of fractions after the decimal point. The n-th row of|φ′_(2D)> is a(f_(i), g_(j))exp[−i2π(f_(i)x+g_(j)y)] assumingj=floor[(n−1)÷M]+1 and i=n−(j−1)×M. A two-dimensional orthogonalfunction system is obtained in this way.

Letting (f₁, g₁) be the coordinates of the first point source on theeffective light source, outputting light from the first point sourceamounts to shifting the pupil function P(f, g) describing the pupil ofthe projection optical system by (f₁, g₁). A pupil function P₁(f, g)that acts on the diffracted light is expressed by P(f+f₁, g+g₁). Theelements of the pupil function P₁(f, g) are arrayed one-dimensionally,as indicated by equation (17). Hence, the pupil function P₁(f g) can beexpressed by a one-dimensional vector:

P ₁=(P ₁(f ₁ ,g ₁)P ₁(f ₂ ,g ₁) . . . P ₁(f _(M) ,g ₁)P ₁(f ₁ ,g ₂) . .. P ₁(f _(M) ,g _(M)))  (18)

For the sake of a detailed expression of P₁, “floor” represents theomission of fractions after the decimal point. The n-th column of P₁ isP₁(f_(i), g_(j)) assuming j=floor[(n−1)÷M]+1 and i=n−(j−1)×M. Atwo-dimensional orthogonal function system is obtained in this way.

Letting (f₂, g₂) be the coordinates of the second point source on theeffective light source, a pupil function P₂(f, g) that acts on the lightemanating from the second point source is expressed by P(f+f₂, g+g₂)obtained by shifting P(f, g) by (f₂, g₂). Like the pupil function P₁(f,g), the pupil function P₂(f, g) can be expressed as a one-dimensionalvector:

P ₂=(P ₂(f ₁ ,g ₁)P ₂(f ₂ g ₁) . . . P ₂(f ₁ ,g ₁)P ₂(f ₁ ,g ₂) . . . P₂(f _(M) /g _(M)))  (19)

If N point sources exist on the effective light source, atwo-dimensional P operator 408 can be defined by:

$\begin{matrix}{P_{2D} = \begin{pmatrix}P_{1} \\P_{2} \\\vdots \\P_{N}\end{pmatrix}} & (20)\end{matrix}$

Using |φ′_(2D)> and P_(2D), a two-dimensional light intensitydistribution I(x, y) formed on the wafer surface can be calculated by:

I(x,y)=(

φ′_(2D) |P _(2D) ⁺ P _(2D)|φ′_(2D)

  (21)

In equation (21), the singular value decomposition of P_(2D) yields:

$\begin{matrix}\begin{matrix}{{I( {x,y} )} = {\langle{\varphi_{2D}^{\prime}{{P_{2D}^{+}P_{2D}}}\varphi_{2D}^{\prime}}\rangle}} \\{= {\langle{\varphi_{2D}^{\prime}{{V^{+}S^{+}W^{+}{WSV}}}\varphi_{2D}^{\prime}}\rangle}} \\{= {\langle{\varphi_{2D}^{\prime}{{V^{+}{SSV}}}\varphi_{2D}^{\prime}}\rangle}} \\{= {S^{2}{\langle{\varphi_{2D}^{\prime}{{V^{+}V}}\varphi_{2D}^{\prime}}\rangle}}} \\{= {S^{2}{\langle{\Phi_{2D}\Phi_{2D}}\rangle}}}\end{matrix} & (22)\end{matrix}$

In this manner, even when a two-dimensional aerial image (lightintensity distribution) is calculated, the elements of the P operator408 are fewer than those of the TCC, and complicated calculation of theP operator 408 is unnecessary. This makes it possible to calculate theaerial image 409 as a light intensity distribution, which is formed onthe wafer surface, in a shorter period of time.

A two-dimensional set of light components diffracted by the mask patterncan be expressed by:

$\begin{matrix}{\psi_{2D} = \begin{pmatrix}{{a( {f_{1},g_{1}} )}^{- {{2\pi}{({{f_{1}x} + {g_{1}y}})}}}} & {{a( {f_{2},g_{1}} )}^{- {{2\pi}{({{f_{2}x} + {g_{1}y}})}}}} & \ldots & {{a( {f_{M},g_{1}} )}^{- {{2\pi}{({{f_{m}x} + {g_{1}y}})}}}} \\{{a( {f_{1},g_{1}} )}^{- {{2\pi}{({{f_{1}x} + {g_{2}y}})}}}} & \ddots & \; & \vdots \\\vdots & \; & \; & \; \\{{a( {f_{1},g_{M}} )}^{- {{2\pi}{({{f_{1}x} + {g_{M}y}})}}}} & \ldots & \; & {{a( {f_{M},g_{M}} )}^{- {{2\pi}{({{f_{M}x} + {g_{M}y}})}}}}\end{pmatrix}} & (23)\end{matrix}$

Referring to equation (23), ψ_(2D) is a matrix of M rows×M columns,which includes M² elements. An operator (stacking operator) Y whichtransforms the matrix of M rows×M columns into that of 1 row×M² columns(i.e., rearranges the elements of ψ_(2D)) in accordance with apredetermined rule is introduced herein. By introducing the stackingoperator Y, a vector |φ′_(2D)> of a two-dimensional diffracted lightdistribution is expressed by:

|φ′_(2D)

=Y[ψ _(2D)]^(T)  (24)

Letting (f₁, g₁) be the coordinates of the first point source on theeffective light source, outputting light from the first point sourceamounts to shifting the pupil of the projection optical system. A pupilfunction P₁(f, g) that acts on the diffracted light is expressed byP(f+f₁, g+g₁). Hence, the pupil function P₁(f, g) can be expressed by:

$\begin{matrix}{P_{1}^{\prime} = \begin{pmatrix}{P_{1}( {f_{1},g_{1}} )} & {P_{1}( {f_{2},g_{1}} )} & \ldots & {P_{1}( {f_{M},g_{1}} )} \\{P_{1}( {f_{1},g_{2}} )} & \ddots & \; & \vdots \\{P_{1}( {f_{1},g_{M}} )} & \ldots & \; & {P_{1}( {f_{M},g_{M}} )}\end{pmatrix}} & (25)\end{matrix}$

Likewise, letting (f₂, g₂) be the coordinates of the second light sourceon the effective light source, a pupil function P₂(f, g) that acts onthe light emanating from the second point source is expressed by P(f+f₂,g+g₂). Like the pupil function P₁(f, g), the pupil function P₂(f, g) canbe expressed by:

$\begin{matrix}{P_{2}^{\prime} = \begin{pmatrix}{P_{2}( {f_{1},g_{1}} )} & {P_{2}( {f_{2},g_{1}} )} & \ldots & {P_{2}( {f_{M},g_{1}} )} \\{P_{2}( {f_{1},g_{2}} )} & \ddots & \; & \vdots \\\vdots & \; & \; & \; \\{P_{2}( {f_{1},g_{M}} )} & \ldots & \; & {P_{2}( {f_{M},g_{M}} )}\end{pmatrix}} & (26)\end{matrix}$

Assume that N point sources exist on the effective light source. Usingthe stacking operator Y, a two-dimensional P operator 408 can beexpressed by:

$\begin{matrix}{P_{2D} = \begin{pmatrix}{Y\lbrack P_{1}^{\prime} \rbrack} \\{Y\lbrack P_{2}^{\prime} \rbrack} \\\vdots \\{Y\lbrack P_{N}^{\prime} \rbrack}\end{pmatrix}} & (27)\end{matrix}$

When an aerial image as a light intensity distribution formed on thewafer surface is calculated from equations (24) and (27), equation (21)or (22) need only be used.

When the TCC is calculated,

T _(2D) =P _(2D) ⁺ P _(2D)  (28)

need only be used.

The P operators 408 according to which the pupil of the projectionoptical system is shifted so as to form point sources in the rowdirection have been explained above, as indicated by equations (20) and(27). However, a P operator 408 according to which the pupil of theprojection optical system is shifted so as to form point sources in thecolumn direction as indicated by:

P ₂ D=(Y[P′ ₁]^(T) Y[P′ ₂]^(T) . . . Y[P′ _(N)]^(T))  (29)

can produce essentially the same effect. Hence, even when the P operator408 is expressed by equation (29), a description of an orthogonalfunction system need only be so adjusted.

FIG. 3 is a flowchart for explaining details of a process of calculatingthe aerial image 409 by the aerial image calculation program 411. Notethat the aerial image calculation program 411 is installed from thestorage medium 70 connected to the medium interface 60, and is stored inthe storage unit 40 via the control unit 20. Also, the aerial imagecalculation program 411 is started up in response to a start-up commandinput from the input unit 50 by the user, and is executed by the controlunit 20.

In step S1002, the control unit 20 determines pieces of aerial imagecalculation information including the effective light source information402, NA information 403, λ information 404, aberration information 405,polarization information 406, resist information 407, and mask data 410.More specifically, the user inputs (selects), to the processingapparatus 1 via the input unit 50, effective light source information“quadrupole illumination”, NA information “0.73”, λ information “248nm”, aberration information “no aberration”, polarization information“non-polarization”, resist information “out of consideration”, and maskdata “contact hole”. Then, the control unit 20 displays, on the displayunit 30, the pieces of aerial image calculation information input(selected) by the user, and determines them. This embodiment willexemplify a case in which the user inputs (selects) the pieces of aerialimage calculation information stored in the storage unit 40. However,the user can input pieces of aerial image calculation information whichare not stored in the storage unit 40.

In step S1004, the control unit 20 calculates the P operator 408. Morespecifically, the control unit 20 receives the pieces of aerial imagecalculation information, which are input (selected) by the user, fromthe storage unit 40. Based on the pieces of aerial image calculationinformation, the control unit 20 calculates the P operator 408 from, forexample, equation (20) or (27). Also, the control unit 20 stores thecalculated P operator 408 in the storage unit 40.

In step S1006, the control unit 20 calculates the aerial image 409. Morespecifically, the control unit 20 calculates the aerial image 409 from,for example, equation (21) or (22) using the P operator 408 and thepieces of aerial image calculation information input (selected) by theuser. Also, the control unit 20 displays the aerial image 409 on thedisplay unit 30, and stores it in the storage unit 40.

In this manner, the process of calculating the aerial image 409 by theaerial image calculation program 411 can calculate the aerial image 409using the P operator 408. In other words, the process of calculating theaerial image 409 by the aerial image calculation program 411 cancalculate the aerial image 409 without calculating the TCC that isnecessary for the SOCS. This makes it possible to simplify the overallcalculation, thus shortening the time taken to calculate the aerialimage 409.

The calculation result of the aerial image 409 obtained by the aerialimage calculation program 411 is analyzed as needed. The analysis of theaerial image includes, for example, the visual confirmation of theaerial image and the confirmation of the NILS (Normalized Intensity LogSlope), the contrast, the defocus characteristic (DOF characteristic),and the degree of matching, with the pattern data 401, of the aerialimage. It is also possible to confirm the action of the aerial image 409on the resist. The analysis of the aerial image can take any form knownto those skilled in the art.

The process of calculating the aerial image 409 by the aerial imagecalculation program 411 can be applied to various forms of themodel-based RET.

The effect of the process of calculating the aerial image 409 by theaerial image calculation program 411, the application of thiscalculation process to the model-based RET, and the like will beexplained in detail in each of the following embodiments.

First Embodiment

The effect of a process of calculating an aerial image 409 by an aerialimage calculation program 411 will be explained in the first embodiment.In the first embodiment, 64-bit Opteron® is used as a CPU whichconstitutes a control unit 20 of a processing apparatus 1, and a memoryof about 10 gigabytes is used as a storage unit 40. The aerial imagecalculation program 411 is generated using MATLAB®, and the time takento calculate the aerial image 409 (calculation time) is compared withthat in the prior art (SOLS).

The first embodiment assumes a case in which an exposure apparatus usesa projection optical system having an NA of 0.73 (corresponding to NAinformation 403), and exposure light having a wavelength of 248 nm(corresponding to λ information 404). In addition, the projectionoptical system is assumed to have no aberration (corresponding toaberration information 405), the illumination light is assumed to benon-polarized (corresponding to polarization information 406), and aresist applied on a wafer is not taken into consideration (correspondingto resist information 407). The effective light source is assumed to usequadrupole illumination, as shown in FIG. 4A. Pattern data (targetpattern) 401 is assumed to include two contact hole patterns. Inaddition, the diameter of each contact hole pattern is assumed to be 120nm, and the centers of the respective contact hole patterns are assumedto be (−120 nm, 0 nm) and (120 nm, 0 nm). Under this assumption, maskdata 410 is as shown in FIG. 4B. In addition, the division number of thepupil of the projection optical system in a normal state is assumed tobe 31, and that in executing Fourier transformation is assumed to be1,024.

The control unit 20 calculates a P operator 408 based on theabove-described pieces of aerial image calculation information. At thistime, the time taken to calculate the P operator 408 is equal to or lessthan 0.1 sec. Also, the control unit 20 performs the singular valuedecomposition of the P operator 408 (i.e., decomposes it intoeigenvalues and eigenfunctions). At this time, the time taken for thesingular value decomposition of the P operator 408 is 0.4 sec. When acomplete aerial image is calculated by adding all the eigenfunctions, anaerial image 409 as shown in FIG. 4C is obtained. The time taken tocalculate the aerial image 409 is about 33.0 sec. Note that the aerialimage 409 shown in FIG. 4C is normalized assuming that its maximum valueis 1.

The aerial image is calculated using the SOCS next. The control unit 20calculates the TCC based on equation (2). At this time, the time takento calculate the TCC is about 1,152 sec. Also, the control unit 20decomposes the TCC into eigenvalues and eigenfunctions based on equation(4) (first calculation step). At this time, the time taken to decomposethe TCC into eigenvalues and eigenfunctions is about 4.9 sec. When acomplete aerial image is calculated by adding all the eigenfunctions(second calculation step), an aerial image as shown in FIG. 4D isobtained. The time taken to calculate the aerial image is about 1,209sec. Note that the aerial image as shown in FIG. 4D is normalizedassuming that its maximum value is 1.

In this manner, the process of calculating the aerial image 409 by theaerial image calculation program 411 can calculate the aerial image in ashorter period of time than in the conventional SOCS. When the aerialimage 409 shown in FIG. 4C and the aerial image shown in FIG. 4D arecompared, they match each other on the order of 1.0×10⁻¹⁵, and thereforea correct simulation result is obtained.

Also, the aerial image calculation program 411 can calculate the TCC ina shorter period of time. The SOCS takes about 1,152 sec to calculatethe TCC, as described above. In contrast, the control unit 20 takes onlyabout 0.9 sec to calculate the TCC based on equation (28) aftercalculating the P operator 408.

Second Embodiment

A process of calculating an aerial image 409 when the aberration of theprojection optical system is taken into consideration or when theillumination light is polarized will be explained in the secondembodiment. A process of calculating an aerial image 409 when theeffective light source has a variation in light intensity (the lightintensity of the light-emitting part is nonuniform) or when thediffraction efficiency of light diffracted by the mask pattern changeswill also be explained in the second embodiment.

When the aberration of the projection optical system is taken intoconsideration, it is only necessary to include the aberration in thepupil function and calculate a P operator 408 using equation (27). Inthis case, each element of the P operator 408 includes a conjugateelement corresponding to the aberration of the projection opticalsystem. The aberration of the projection optical system can be includedin P′_(i) in equation (27) in this way.

Pieces of aerial image calculation information other than aberrationinformation 405 are assumed to be the same as in the first embodiment.An aberration of 50 mλ is substituted into the seventh term (low-ordercoma aberration) of the Fringe Zernike polynomial as the aberrationinformation 405. FIG. 5 shows the result of calculating the aerial image409 using the P operator 408 for mask data 410 shown in FIG. 4B, inaccordance with an aerial image calculation program 411. Referring toFIG. 5, two contact hole patterns exhibit aerial image elements havingdifferent sizes, that is, the left contact pattern exhibits a largeraerial image element. This is because a low-order coma aberration is setas the aberration of the projection optical system.

Since the defocus is one type of wavefront aberration, it can beincluded in the process of calculating the aerial image 409 by theaerial image calculation program 411. When a resist applied on a waferis assumed as a parallel plate, the resist can be said to generatespherical aberration. It is therefore possible to include resist-relatedaberration in the process of calculating the aerial image 409 by theaerial image calculation program 411.

When the illumination light is polarized, it is only necessary tothree-dimensionally express the polarization by making σ=1 for theeffective light source correspond to the NA of the projection opticalsystem. More specifically, the pupil function need only be multiplied byfactors associated with polarization. The factors associated withpolarization include a factor which produces an effect of allowingx-polarized light remain as x-polarized, that which produces an effectof turning x-polarized light into y-polarized light, that which producesan effect of turning x-polarized light into z-polarized light, thatwhich produces an effect of turning y-polarized light into x-polarizedlight, that which produces an effect of allowing y-polarized lightremain as y-polarized, and that which produces an effect of turningy-polarized light into z-polarized light. Accordingly, three P operatorsfor x-polarization, y-polarization, and z-polarization are derived.

A light intensity distribution I(x, y) formed on the wafer surface canbe calculated by:

I(x,y)=

φ′_(2D) |P _(x) +P _(x) +P _(y) ⁺ P _(y) +P _(z) ⁺ P _(z)|φ′_(2D)

  (30)

where P_(X), P_(y), and P_(z) are the P operators for x-polarization,y-polarization, and z-polarization, respectively.

Pieces of aerial image calculation information other than polarizationinformation 406 are assumed to be the same as in the first embodiment.That all the point sources are polarized in the x direction is set asthe polarization information 406. FIG. 6 shows the result of calculatingthe aerial image 409 using the P operators P_(x), P_(y), and P_(z) forx-polarization, y-polarization, and z-polarization, respectively, inaccordance with the aerial image calculation program 411. When FIGS. 6and 4C are compared, the aerial image 409 when the illumination ispolarized is more blurred than that when the illumination light isnon-polarized.

A P operator is defined by:

$\begin{matrix}{P_{pol} = \begin{pmatrix}P_{x} \\P_{y} \\P_{z}\end{pmatrix}} & (31)\end{matrix}$

where P_(pol) is a P operator including the effect of polarization.Since the elements of P_(pol) are fewer than those of the TCC, the useof P_(pol) allows calculating the aerial image 409 in a shorter periodof time than in the SOCS.

One important conclusion can be derived herein. That is, one P operatorhas conventionally been defined by one matrix, but a plurality of Poperators may be defined by different matrices according to the presentinvention. For example, one P operator P_(pol) is defined by takingaccount of polarization in the second embodiment, but three P operatorscan be defined for respective polarized light components. A plurality ofP operators may also be defined by dividing the effective light sourceinto a plurality of regions.

If the effective light source has a variation in light intensity (thelight intensity of the light-emitting part is nonuniform), the intensityof each point source need only be included in the P operator. Forexample, when the intensity of the i-th point source is S_(i), the Poperator is defined by:

$\begin{matrix}{P_{2D} = \begin{pmatrix}{\sqrt{S_{1}}{Y\lbrack P_{1}^{\prime} \rbrack}} \\{\sqrt{S_{2}}{Y\lbrack P_{2}^{\prime} \rbrack}} \\\vdots \\{\sqrt{S_{N}}{Y\lbrack P_{N}^{\prime} \rbrack}}\end{pmatrix}} & (32)\end{matrix}$

The diffracted light distribution upon normal incidence often differsfrom that upon oblique incidence (i.e., the diffraction efficiency oftenchanges) along with advance in the micropatterning of the mask pattern.In this case, equation (27) need only be rewritten as:

$\begin{matrix}{P_{2D} = \begin{pmatrix}{Y\lbrack P_{1}^{''} \rbrack} \\{Y\lbrack P_{2}^{''} \rbrack} \\\vdots \\{Y\lbrack P_{N}^{''} \rbrack}\end{pmatrix}} & (33)\end{matrix}$

Note that P″_(i) in equation (33) includes the diffraction efficiencywhen light emanating from the i-th point source obliquely enters thewafer surface.

Third Embodiment

The third embodiment will exemplify a case in which a process ofcalculating an aerial image 409 by an aerial image calculation program411 is applied to the model-based RET. The OPC (Optical ProximityCorrection) is known as a basic approach to the RET.

Pieces of aerial image calculation information other than mask data 410are assumed to be the same as in the first embodiment. In the thirdembodiment, five bars each having a width of 120 nm and a length of 840nm, as shown in FIG. 7A, are assumed as the mask data 410. FIG. 7B showsan aerial image 409 calculated by the aerial image calculation program411 in this case. When FIGS. 7A and 7B are compared, the mask data 410differs from the aerial image 409 calculated by the aerial imagecalculation program 411. To solve this problem, based on the OPC, themask data 410 is changed so that the aerial image 409 calculated by theaerial image calculation program 411 (i.e., a pattern transferred byexposure) becomes close to pattern data 401.

To determine optimal mask data 410 in the OPC, a loop for calculatingthe aerial image 409 by changing the mask data 410 must be repeateduntil the difference between the aerial image 409 and the pattern data401 decreases sufficiently. For this reason, when a long time is takento calculate the aerial image 409, a long time is, in turn, taken todetermine optimal mask data 410. However, since the aerial imagecalculation program 411 can calculate the aerial image 409 in a shorterperiod of time, it is suitable for the OPC.

More specifically, to determine optimal mask data 410, the control unit20 repeats the process of calculating the aerial image 409 by the aerialimage calculation program 411 for the above-described five bars, therebychanging the mask data 410. With this operation, final mask data 410 isobtained, in which the leftmost and rightmost bars each have a width of134 nm and a length of 968 nm, the second bars from the left and righteach have a width of 127 nm and a length of 930 nm, and the central barhas a width of 120 nm and a length of 929 nm. FIG. 7C shows an aerialimage 409 calculated by the aerial image calculation program 411 usingthe mask data 410. When FIGS. 7B and 7C are compared, the aerial image409 shown in FIG. 7C is closer to the pattern data 401 than that 409shown in FIG. 7B.

In this manner, applying the process of calculating the aerial image 409by the aerial image calculation program 411 to the OPC makes it possibleto generate mask data 410 in a shorter period of time.

Fourth Embodiment

A method of calculating an aerial image 409 in a shorter period of timethan in the previous embodiments in a process of calculating the aerialimage 409 by an aerial image calculation program 411 will be explainedin the fourth embodiment.

When a two-dimensional aerial image 409 is calculated, a P operator 408can be expressed by equation (27), as described above. Letting L be thepupil division number, and N be the number of point sources, the Poperator is defined by a matrix of N rows and (2L)² columns. Since therows of the P operator are independent of each other, the rank of the Poperator is N. In other words, the singular value decomposition of the Poperator yields N eigenvalues and N eigenfunctions. N eigenvalues and Neigenfunctions are therefore necessary to calculate a complete aerialimage 409. In practice, however, N eigenvalues and N eigenfunctions neednot be used, as will be described later.

Pieces of aerial image calculation information other than mask data 410are assumed to be the same as in the first embodiment. In the fourthembodiment, five bars each having a width of 120 nm and a length of 840nm, as shown in FIG. 7A, are assumed as the mask data 410.

The effective light source shown in FIG. 4A has 92 point sources fornumerical calculation. This means that there are 92 eigenvalues and 92eigenfunctions. The eigenfunctions corresponding to the eigenvalues arerearranged in descending order of the squares of the eigenvalues.

FIG. 8A is a graph obtained by plotting the squares of the eigenvalues,which are normalized assuming that the square of the largest eigenvalueis 1. As described above, there are 92 eigenvalues. Letting i be theeigenvalue number, that is, the i-th eigenvalue, the square of the i-theigenvalue is 0.01 or less if i is 10 or more, as shown in FIG. 8A.Also, the square of the i-th eigenvalue is 0.001 or less if i is 49 ormore. In this manner, the square of the i-th eigenvalue rapidlydecreases along with an increase in the eigenvalue number i.

Let E be the sum of the squares of all the eigenvalues, and Ei be thesum of the squares of the first to i-th eigenvalues herein. WhenE_(i)/E=1, a complete aerial image 409 can be calculated. When the lightintensity is set such that the central bar in a section y=0 has a linewidth of 120 nm in the complete aerial image 409, the leftmost bar has aline width of 98.44 nm.

The square of the eigenvalue rapidly decreases along with an increase inthe eigenvalue number i, and the special frequency of the eigenfunctionincreases along with an increase in the eigenvalue number i. Accordingto this principle, as the eigenvalue number i increases, thecontribution of the i-th eigenvalue to the formation of an aerial imagedecreases. FIG. 8B is a graph showing the difference between a completeaerial image and an approximate aerial image (i.e., an aerial imagecalculated from some eigenvalues and eigenfunctions). In FIG. 8B, theabscissa indicates the eigenvalue number i, and the ordinate indicatesthe line width of the leftmost bar when an aerial image is calculatedusing the first to i-th eigenfunctions and eigenvalues. Referring toFIG. 8B, when E_(i)/E=0.96 or more (i.e., the eigenvalue number i is 14or more), the difference between the complete aerial image and theapproximate aerial image is 0.1 nm or less. Hence, an aerial imagealmost identical to the complete aerial image can be calculated usingnot 92 eigenvalues and 92 eigenfunctions but 14 eigenvalues and 14eigenfunctions. This makes it possible to reduce the time taken tocalculate the aerial image by about 85%.

A required accuracy of the aerial image differs depending on theevaluation target. Through the examinations of various cases, theinventor of the present invention found that when E_(i)/E is 0.96 ormore, no problems are posed from the viewpoint of practical application.Furthermore, the inventor of the present invention found that whenE_(i)/E is 0.98 or more, no problems are posed for almost all theevaluation targets.

In this manner, the aerial image 409 can be calculated in a shorterperiod of time than in the previous embodiments by adjusting E_(i)/E inaccordance with the evaluation target in the process of calculating theaerial image 409 by the aerial image calculation program 411.

Another method of calculating the aerial image 409 in a shorter periodof time than in the previous embodiments includes a method ofcompressing the P operator 408. For example, consider one-dimensionalimaging. If all the components of the j-th column of the P operator 408are 0, the j-th column of the P operator 408 is not necessary at all.Then, the P operator 408 can be compressed by eliminating columns ineach of which all the components are 0 in the P operator 408 asindicated by:

$\begin{matrix}{{\begin{pmatrix}0 & 0 & 1 & 1 & 1 & 0 & 0 \\0 & 1 & 1 & 1 & 0 & 0 & 0\end{pmatrix}\begin{pmatrix}^{{- 2}\pi \; v_{1}x} \\^{{- 2}\pi \; v_{2}x} \\^{{- 2}\pi \; v_{3}x} \\^{{- 2}\pi \; v_{4}x} \\^{{- 2}\pi \; v_{5}x} \\^{{- 2}\pi \; v_{6}x} \\^{{- 2}\pi \; v_{7}x}\end{pmatrix}} = {\begin{pmatrix}0 & 1 & 1 & 1 \\1 & 1 & 1 & 0\end{pmatrix}\begin{pmatrix}^{{- 2}\pi \; v_{2}x} \\^{{- 2}\pi \; v_{3}x} \\^{{- 2}\pi \; v_{4}x} \\^{{- 2}\pi \; v_{5}x}\end{pmatrix}}} & (34)\end{matrix}$

Referring to equation (34), a P operator 408 of two rows and sevencolumns is compressed into that of two rows and four columns.

Likewise, the P operator 408 can be compressed even in two-dimensionalimaging. More specifically, the P operator 408 can be compressed byeliminating columns in each of which all the components are 0 in the Poperator 408. The use of the P operator 408 compressed in this wayallows the singular value decomposition in a shorter period of time thanin the previous embodiments. This makes it possible to calculate theaerial image 409 in a shorter period of time than in the previousembodiments.

A case in which the P operator 408 is compressed will be exemplified indetail. The effective light source is assumed to use quadrupoleillumination and include 712 point sources, as shown in FIG. 9A. FIG. 9Bshows a P operator 408 which is normally calculated in accordance withthe aerial image calculation program 411 (i.e., uncompressed). The Poperator 408 shown in FIG. 9B is a matrix of 712 rows and 16,129columns, in which the white portion corresponds to 1, and the blackportion corresponds to 0. FIG. 9C shows a P operator 408 compressedthrough a predetermined process. The P operator 408 shown in FIG. 9C isa matrix of 712 rows and 10,641 columns, in which the white portioncorresponds to 1, and the black portion corresponds to 0.

The time taken for the singular value decomposition of the P operator408 shown in FIG. 9B is about 34.0 sec. In contrast, the time taken forthe singular value decomposition of the P operator 408 shown in FIG. 9Cis about 24.3 sec. In this manner, the compression of the P operator 408improves the speed at which the singular value decomposition isperformed for the P operator 408.

Fifth Embodiment

A method of generating mask data 410 using the model-based RET whichuses a P operator 408, particularly a process of calculating an aerialimage 409 by an aerial image calculation program 411 will be explainedin the fifth embodiment. This method generates mask data 410 byinserting assist (auxiliary) patterns in the pattern to be transferredby exposure.

FIG. 10 is a schematic block diagram showing the configuration of aprocessing apparatus 1 according to the fifth embodiment. The processingapparatus 1 shown in FIG. 10 has basically the same configuration asthat of the processing apparatus 1 shown in FIG. 1, but a storage unit40 additionally stores a mask function 412, P map 413, and maskgeneration program 414. Effective light source information 402, NAinformation 403, λ information 404, aberration information 405,polarization information 406, resist information 407, the mask data 410,and the mask function 412 will be collectively referred to as pieces ofP map calculation information hereinafter.

The mask function 412 is a parameter for generating a P map 413 (to bedescribed later), and is pattern data 401 itself or the one obtained bytransforming the pattern data 401 in accordance with a predeterminedrule.

The P map 413 is a partial coherent map obtained by multiplying theeigenfunctions of the P operator 408 by the diffracted lightdistribution, and Fourier-transforming or adding the products.

The difference between a process of calculating the P map 413 and thatof calculating the aerial image 409 will be explained herein. Theprocess of calculating the aerial image 409 multiplies theeigenfunctions of the P operator 408 by the diffracted lightdistribution on the mask data 410, Fourier-transforms the products, andcalculates the squares of the absolute values of the Fourier transforms.The squares of the absolute values are then multiplied by the squares ofthe absolute values of their corresponding eigenvalues, and the productsare added. With this operation, the aerial image 409 is calculated. Incontrast, the P map 413 is calculated by multiplying the eigenfunctionsof the P operator 408 by the diffracted light distribution on the maskdata 410, Fourier-transforming the products, multiplying the Fouriertransforms by their corresponding eigenvalues, and adding the products.Therefore, the aerial image 409 always takes positive values, but the Pmap 413 does not always take positive values. The P map 413 means a map(function) which represents, when a plurality of pattern elements areinserted on the object plane of the projection optical system, theinfluence they inflict on each other.

The mask generation program 414 is a program for generating mask data410 based on the P map 413.

A process of generating the mask data 410 by the mask generation program414 will be explained below by paying particular attention to theinsertion of assist patterns.

The fifth embodiment assumes a case in which an exposure apparatus usesa projection optical system having an NA of 0.73 (corresponding to theNA information 403), and exposure light having a wavelength of 248 nm(corresponding to the λ information 404). In addition, the projectionoptical system is assumed to have no aberration (corresponding to theaberration information 405), the illumination light is assumed to benon-polarized (corresponding to the polarization information 406), and aresist applied on a wafer is not taken into consideration (correspondingto the resist information 407). The effective light source(corresponding to the effective light source information 402) is assumedto use quadrupole illumination, as shown in FIG. 11A. Pattern data(target pattern) 401 is assumed to be an isolated contact hole pattern,which has a side length of 120 nm, as shown in FIG. 11B.

The fifth embodiment also assumes a so-called dark field with a clearaperture. In this case, a pattern is formed on a resist portionirradiated with the exposure light.

Since various values can be set to the wavelength λ of the exposurelight, and the numerical aperture NA of the projection optical system inthe exposure apparatus, the mask pattern size is preferably normalizedby (π/NA). For example, if λ=248 nm and NA=0.73, a pattern having a sizeof 100 nm is normalized to 0.29 by the above-described method. Thisnormalization will be referred to as k1 conversion hereinafter.

The k1 conversion value of an isolated contact hole pattern having adiameter of 120 nm is 0.35. If the k1 conversion value is 0.5 or less, asinusoidal aerial image is obtained. To make the best use of thefeatures of a sinusoidal wave, assist patterns have conventionally beeninserted at a cycle half the diameter of the isolated contact holepattern. For example, if the center of the isolated contact hole patternas a desired pattern lies at (0, 0), assist patterns have been insertedat eight positions, that is, (±240, 0), (0, ±240), (240, ±240), and(−240, ±240).

In the mask generation program 414, first, the mask function 412 is setas the target pattern itself, that is, an isolated contact hole patternhaving a diameter of 120 nm.

As has been described in the fourth embodiment, as the eigenvalue of theP operator 408 increases, the contribution of the eigenfunction of the Poperator 408 to the formation of the aerial image 409 increases. Toattain this state, the eigenvalues of the P operator 408 are rearrangedin descending order of the squares of the eigenvalues. An eigenfunctioncorresponding to the i-th eigenvalue rearranged in this way will bereferred to as the i-th eigenfunction hereinafter.

The first eigenfunction of the P operator 408 makes a largestcontribution to the formation of the aerial image 409. For this reason,only the first eigenfunction of the P operator 408 is considered. Thefirst eigenfunction of the P operator 408 is multiplied by thediffracted light distribution of the mask function 412, and the productis Fourier-transformed. FIG. 11C shows a P map 413 calculated in thisway.

In FIG. 11C, the values in regions AR1 to AR8 surrounded by the whitedotted lines are relatively large. In other words, light componentsdiffracted by the regions AR1 to AR8 interfere with that diffracted bythe target pattern, thereby improving the image intensity. Hence, whenopening patterns are inserted in the regions AR1 to AR8 surrounded bythe white dotted lines, the image intensity at the position (0, 0)increases.

Assist patterns HP1 to HP8 are inserted in the regions AR1 to AR8surrounded by the white dotted lines, as shown in FIG. 11D. The originalpurpose is to transfer by exposure an isolated contact hole patternhaving its center at (0, 0) onto the wafer surface, as described above.When the P map 413 is analyzed, a position at which the P map 413 takesa peak value is (0, 0). The assist patterns HP1 to HP8 are set such thatthe center of a main pattern SP having the same size as that of thepattern data 401 becomes (0, 0). A mask is then fabricated by using themask pattern shown in FIG. 11D as the mask data 410. With thisoperation, light components diffracted by the assist patterns HP1 to HP8act on that diffracted by the main pattern SP. This makes it possible totransfer the isolated contact hole pattern as the target pattern withhigh accuracy, thus improving the resolving performance.

FIG. 12 shows the result of a comparison of the imaging performances ofa mask having no assist patterns, that in which assist patterns areinserted according to the prior art, and that in which assist patternsare inserted according to the fifth embodiment (i.e., by the maskgeneration program 414). In FIG. 12, the abscissa indicates the defocusamount, and the ordinate indicates the diameter of the contact holepattern (CD). The imaging performance of each mask is evaluated based ona change in the diameter of the isolated contact hole pattern (CD) withrespect to a change in defocus. The size of each assist pattern in theprior art is 90 nm×90 nm. The size of each assist pattern in the fifthembodiment will be described later.

The imaging performance of the mask having no assist patterns, and thatof the mask in which assist patterns are inserted according to the priorart will be compared with reference to FIG. 12. In this case, a changein the diameter of the isolated contact hole pattern with respect to achange in defocus in the mask in which assist patterns are insertedaccording to the prior art is significantly smaller than that in themask having no assist patterns. In other words, the mask in which assistpatterns are inserted according to the prior art exhibits an imagingcharacteristic better than that of the mask having no assist patterns.

Likewise, the imaging performance of the mask in which assist patternsare inserted according to the prior art, and that of the mask in whichassist patterns are inserted according to the fifth embodiment will becompared. Referring to FIG. 12, the mask in which assist patterns areinserted according to the fifth embodiment exhibits an imagingcharacteristic better than that of the mask in which assist patterns areinserted according to the prior art.

The P map 413 shown in FIG. 11C will be explained in detail. On the Pmap 413 shown in FIG. 11C, the first position which exhibits a valuemore than a predetermined threshold and takes a peak value is detected.The position which takes a peak value means a position at which a valueobtained by differentiating the P map 413 with respect to the positionis zero. The first position is a vector, which includes pieces ofinformation on the distance and direction. In the fifth embodiment, thefirst positions are eight positions, that is, (±285, 0), (0, ±285),(±320, 320), and (±320, −320). Note that in FIG. 11C the P map 413 isnormalized assuming that its maximum value is 1, and the threshold is0.03. As the first positions are calculated, assist patterns areinserted on the light intensity distribution on the P map 413 asfaithfully as possible. In the fifth embodiment, assist patterns ofrotationally symmetrical rectangles each having a size of 70 nm×120 nmare inserted (arranged) at the eight positions.

It is difficult to calculate the peak position by numerical calculation.Using the fact that the peak position and the barycentric position arenearly the same, the barycenter of a region which exhibits a value morethan the predetermined threshold on the P map 413 may be calculated andset as the first position.

For example, FIG. 11E shows a map obtained by setting each region whichexhibits a value equal to or more than the threshold of 0.03 as 1, andsetting each region which exhibits a value less than the threshold of0.03 as 0 on the P map 413 shown in FIG. 11C. In FIG. 11E, a region SRcorresponds to the main pattern SP as the isolated contact hole pattern(desired pattern). Regions HR1 to HR8 correspond to regions (i.e., theregions AR1 to AR8) to insert (arrange) the assist patterns. Hence, theassist patterns need only be set by calculating the barycenters of theregions HR1 to HR8.

Sixth Embodiment

A method of generating mask data 410 when pattern data (target pattern)401 is a pattern formed from n contact hole patterns will be explainedin the sixth embodiment.

As has been described in the fifth embodiment, generating mask data 410using a P map 413 improves the imaging performance of a mask having anisolated contact hole pattern. Likewise, generating mask data 410 usinga P map 413 also makes it possible to improve the imaging performance ofa mask having a pattern formed from n contact hole patterns.

Pieces of P map calculation information other than a mask function 412are assumed to be the same as in the fifth embodiment. The pattern data(target pattern) 401 is assumed to be a pattern including three 120-nmsquare contact hole patterns, as shown in FIG. 13A. The centers of thethree contact hole patterns are (0, 0), (320, 320), and (640, −350). Themask function 412 is the target patter itself, that is, a patternincluding three contact hole patterns each having a diameter of 120 nm,as in the fifth embodiment.

The first eigenfunction of a P operator 408 makes a largest contributionto the formation of an aerial image 409. For this reason, only the firsteigenfunction of the P operator 408 is considered. The firsteigenfunction of the P operator 408 is multiplied by the diffractedlight distribution of the mask function 412, and the product isFourier-transformed. FIG. 13B shows a P map 413 calculated in this way.

In FIG. 13B, a region surrounded by each white dotted line exhibits avalue equal to or more than a certain threshold (0.025 in the fifthembodiment) and corresponds to a peak position. Assist patterns needonly be inserted (arranged) in the regions surrounded by the whitedotted lines shown in FIG. 13B.

How to determine the main pattern corresponding to the pattern data 401will be explained next. Attention is paid to pattern data having itscenter at the position (0, 0). When the P map 413 is analyzed, it has apeak at a position shifted from (0, 0) by (δx, δy). When a 120-nm squaremain pattern having its center at the position (0, 0) is arranged, it istransferred by exposure while being shifted by (δx, δy) due to theoptical proximity effect.

The positional shift can be canceled by arranging a main pattern havingits center at the position (−δx, −δy). Likewise, when the main patternsare contact hole patterns having their centers at the positions (320,320) and (640, −350), they are similarly inserted (arranged) atpositions different from that of the pattern data 401.

The analysis of the P map 413 also makes it possible to predict thedegree of deformation of the main pattern. It is therefore possible todetermine the shape of the main pattern based on its deformation.

After the main pattern is determined as a pattern represented by thepattern data 401 itself based on the P map 413, the positional shift andshape of the main pattern may be corrected by adopting the OPC.

Assist patterns are often arranged too close to each other, depending onthe arrangement of the contact hole patterns. In this case, one assistpattern need only be inserted (arranged) near the adjacent assistpatterns. If a certain assist pattern is close to a desired pattern, itmust be removed.

Seventh Embodiment

The application target of a mask generation program 414 is notparticularly limited to a mask having a square contact hole pattern, andit can be applied to a mask having a rectangular contact hole pattern orline pattern. The seventh embodiment will exemplify a case in which maskdata 410 having an isolated line pattern is generated using the maskgeneration program 414.

Pieces of P map calculation information other than effective lightsource information 402 and a mask function 412 are assumed to be thesame as in the fifth embodiment. The effective light source(corresponding to the effective light source information 402) is assumedto use dipole illumination, as shown in FIG. 14A. Pattern data (targetpattern) 401 is assumed to be an isolated line pattern, which has awidth of 120 nm.

The seventh embodiment also assumes a so-called clear field with anopaque feature, in which a pattern is formed on a resist portion whichis irradiated with the exposure light and exhibits a value equal to orless than a certain threshold.

The mask function 412 is set as the target pattern itself, that is, a120-nm isolated line pattern first. In general, when a line pattern istransferred by exposure, the resist remains unremoved only in the lineportion. Hence, the mask function 412 represents a mask which exhibits abackground transmittance of 100% and has a light-shielding portionformed from a 120-nm isolated line pattern.

FIG. 14B shows a P map 413 calculated from the above-described pieces ofP map calculation information. When assist patterns are inserted(arranged) in regions each of which exhibits a value less than apredetermined threshold and corresponds to a peak position on the P map413 shown in FIG. 14B, imaging performance of the mask improves.

The P map 413 shown in FIG. 14B has peaks at positions about 290 nm fromthe center of the isolated line pattern. The use of a mask in whichassist patterns are arranged at positions 290 nm from the center of theisolated line pattern improves the imaging performance, as shown in FIG.14C. In FIG. 14C, reference symbol d indicates the distance from thecenter of the isolated line pattern, which is 290 nm in the seventhembodiment.

It is also possible to generate mask data 410 for a mask having anisolated line pattern even by inserting (arranging) assist patterns inthe following way.

First, a P map 413 calculated as a dark field with a clear aperture isset as PM₁(x, y). Note that the P map 413 is normalized assuming thatthe maximum value of PM₁(x, y) is 1. Next, PM₂(x, y) as a new P map 413is calculated by setting PM₂(x, y)=1−PM₁(x, y). Assist patterns areinserted (arranged) in regions each of which exhibits a value less thana predetermined threshold and corresponds to a peak position (orbarycentric position) on PM₂(x, y) calculated in this way, therebygenerating mask data 410. Assist patterns which form a clear field withan opaque feature may be inserted (arranged) in regions each of whichexhibits a value more than a predetermined threshold and corresponds toa peak position (or barycentric position).

Eighth Embodiment

A mask function 412 will be explained in detail in the eighthembodiment.

Pieces of P map calculation information are assumed to be the same as inthe fifth embodiment. Pattern data (target pattern) 401 is assumed to bea 120-nm square isolated contact hole pattern.

FIGS. 15A and 15B show a P map 413 calculated by setting the maskfunction 412 as the target pattern itself. FIG. 15A shows the P map 413itself. FIG. 15B shows a map obtained by setting each position having apositive value as 1, and setting each position having a negative valueas −1 on the P map 413 shown in FIG. 15A.

FIGS. 15C and 15D show a P map 413 calculated by setting the maskfunction 412 as a 60-nm square isolated contact hole pattern. FIG. 15Cshows the P map 413 itself. FIG. 15D shows a map obtained by settingeach position having a positive value as 1, and setting each positionhaving a negative value as −1 on the P map 413 shown in FIG. 15C.

FIGS. 15E and 15F show a P map 413 calculated by setting the maskfunction 412 as a 1-nm square isolated contact hole pattern. FIG. 15Eshows the P map 413 itself. FIG. 15F shows a map obtained by settingeach position having a positive value as 1, and setting each positionhaving a negative value as −1 on the P map 413 shown in FIG. 15E.

When a relatively small pattern is set as the mask function 412, assistpatterns are inserted (arranged) such that light converges on the smallpattern, resulting in an increase in exposure margin. However, as can beunderstood from FIGS. 15A to 15F, the mask shape is complicated in thiscase. In contrast, when a relatively large pattern is set as the maskfunction 412, the mask shape is simple. According to the examinations ofvarious cases by the inventor of the present invention, a pattern havinga size equal to or smaller than the target pattern is desirably set asthe mask function 412.

To simplify the calculation of the P map 413, the mask function 412 needonly be set by approximating a contact hole pattern by a point (e.g., a1-nm contact hole pattern), and approximating a line pattern by a line(e.g., a pattern having a width of 1 nm). If a rectangular contact holepattern is used, a line extending in the longitudinal direction (e.g., apattern having a length equal to that of the rectangular contact holepattern in the longitudinal direction, and a width of 1 nm) need only beset as the mask function 412.

For example, in the fifth embodiment, a 1-nm isolated contact holepattern need only be set as the mask function 412. In the sixthembodiment, a pattern including three 1-nm contact hole patterns needonly be set as the mask function 412. In the seventh embodiment, a linepattern having a width of 1 nm need only be set as the mask function412.

The mask function 412 is desirably a pattern having a size equal to orsmaller than the target pattern, as described above. Accordingly, thereduction magnification may be preset to 0 (exclusive) to 1 (inclusive),and the product of the reduction magnification and the originaldimension of the target pattern may be set as the mask function 412.

For example, if the reduction magnification is set to 0.75, a 90 nm (120nm×0.75) isolated contact hole pattern need only be set as the maskfunction 412 in the fifth embodiment. A line pattern having a width of90 nm need only be set as the mask function 412 in the seventhembodiment. Note that the fifth to seventh embodiments each exemplify acase in which the reduction magnification is set to 1.

It is generally difficult to resolve a line pattern and rectangularpattern in the widthwise direction, so attention must be paid to thepattern resolution in the widthwise direction. For this reason, the Pmap 413 may be calculated by setting, as the mask function 412, theproduct of the reduction magnification and the original dimension of thetarget pattern in the widthwise direction.

Ninth Embodiment

Assist patterns inserted (arranged) in regions (positions) each having anegative value on a P map 413 will be explained in the ninth embodiment.

The P map 413 includes regions having negative values. This means thatthere are regions which cancel the formation of an aerial image on the Pmap 413.

The effect of canceling the formation of an aerial image can beinterpreted as inverting the phase of light (i.e., setting light to be180° out of phase). Hence, the imaging performance of the mask can beimproved by inserting (arranging) an assist pattern in each regionhaving a negative value on the P map 413 such that light transmittedthrough a desired pattern becomes 180° out of phase with thattransmitted through the assist pattern.

Pieces of P map calculation information are assumed to be the same as inthe fifth embodiment. Pattern data (target pattern) 401 is assumed to bean isolated contact hole pattern, which has a diameter of 120 nm.

As has been described in the fifth embodiment, the P map 413 shown inFIG. 11C is calculated from the above-described pieces of P mapcalculation information. When assist patterns in phase (0° out of phase)with a desired pattern are inserted (arranged) in regions AR1 to AR8surrounded by the white dotted lines shown in FIG. 11C, the imagingperformance of the mask improves. Note that regions AR9 to AR12surrounded by the white dotted lines have relatively large negative peakvalues on the P map 413 shown in FIG. 11C, as shown in FIG. 16A. On theP map 413 shown in FIG. 16A, the centers of the regions AR9 to AR12surrounded by the white dotted lines lie at four positions, that is,(±225, 225) and (±225, −225). Mask data 410 shown in FIG. 16B isgenerated by inserting (arranging) assist patterns AP1 to AP4, that are180° out of phase with the desired pattern, in the regions AR9 to AR12surrounded by the white dotted lines. In FIG. 16B, the assist patternsAP1 to AP4 each are 180° out of phase with the desired pattern and havea size of 90 nm×90 nm.

FIG. 17 is a graph showing the result of a comparison of the imagingperformances of a mask based on the mask data 410 shown in FIG. 11D(i.e., according to the fifth embodiment), and that based on the maskdata 410 shown in FIG. 16B (i.e., according to the ninth embodiment). InFIG. 17, the abscissa indicates the defocus amount, and the ordinateindicates the diameter of the isolated contact hole pattern (CD). Theimaging performance of each mask is evaluated based on a change in thediameter of the isolated contact hole pattern (CD) with respect to achange in defocus. Referring to FIG. 17, the mask based on the mask data410 shown in FIG. 16B exhibits an imaging performance better than thatof the mask based on the mask data 410 shown in FIG. 11D.

In this manner, the mask imaging performance can be improved byinserting (arranging) assist patterns, that are 180° out of phase withthe desired pattern, in regions having negative values on the P map 413.Hence, it is only necessary to insert assist patterns, that are in phasewith the desired pattern, in regions each of which exhibits a value morethan a positive threshold and corresponds to a peak position, and toinsert assist patterns, that are 180° out of phase with the desiredpattern, in regions each of which exhibits a value less than a negativethreshold and corresponds to a peak position. In the ninth embodiment,the positive threshold is 0.03, and the negative threshold is −0.018.

Tenth Embodiment

As has been described in the fifth and ninth embodiments, the maskimaging performance can be improved by generating mask data 410 based ona P map 413. However, when assist patterns are inserted (arranged)faithfully to the P map 413, the mask shape is complicated. The currentmask fabrication technique can fabricate a mask based on the mask data410 shown in FIG. 11D, and that based on the mask data 410 shown in FIG.16B. Even so, it is very useful to reduce the load imposed on the maskfabrication.

To reduce the load imposed on the mask fabrication, assist patternsalmost similar to the pattern to be transferred by exposure need only beinserted in regions each of which exhibits a value more than apredetermined threshold and corresponds to a peak position on the P map413. Because the P map 413 has a most conspicuous feature in specifyingpositions to insert (arrange) assist patterns, a change in the shape ofeach assist pattern has little influence on the mask imagingperformance.

Pieces of P map calculation information are assumed to be the same as inthe fifth embodiment. Pattern data (target pattern) 401 is assumed to bean isolated contact hole pattern, which has a diameter of 120 nm.

As has been described in the fifth embodiment, regions each of whichexhibits a value equal to or more than the positive threshold andcorresponds to a peak position on the P map 413 calculated from theabove-described pieces of P map calculation information lie at eightpositions, that is, (±285, 0), (0, ±285), (±320, 320), and (±320, −320).Assist patterns that are in phase with the pattern to be transferred byexposure are inserted at these eight positions. Note that each assistpattern is assumed to be similar to the isolated contact pattern as adesired pattern, and have a size of 90 nm×90 nm.

As has been described in the ninth embodiment, assist patterns, that are180° out of phase with light diffracted by the pattern to be transferredby exposure, are inserted at four positions, that is, (±225, 225) and(±225, −225). Note that each assist pattern is assumed to be similar tothe isolated contact pattern as a desired pattern to be transferred byexposure, and have a size of 90 nm×90 nm.

FIG. 18 shows mask data 410 generated in this way. In FIG. 18, assistpatterns AP5 to AP12 are in phase with the pattern to be transferred byexposure and are inserted (arranged) at (±285, 0), (0, ±285), (±320,320), and (±320, −320). Assist patterns AP13 to AP16 are 180° out ofphase with the pattern to be transferred by exposure and are inserted at(±225, 225) and (±225, −225). Since the assist patterns AP13 to AP16each have a square shape (i.e., are similar to the desired pattern) on amask based on the mask data 410 shown in FIG. 18, this mask can befabricated more easily than a mask based on the mask data 410 shown inFIG. 16B.

FIG. 19 is a graph showing the result of a comparison of the imagingperformances of a mask based on the mask data 410 shown in FIG. 16B(i.e., according to the ninth embodiment), and that based on the maskdata shown in FIG. 18 (i.e., according to the 10th embodiment).Referring to FIG. 19, there is little difference in imaging performancebetween the mask based on the mask data 410 shown in FIG. 16B and thatbased on the mask data 410 shown in FIG. 18. In this manner, the loadimposed on the mask fabrication can be reduced by inserting (arranging)assist patterns that are almost similar to the pattern to be transferredby exposure in regions each of which exhibits a value more than apredetermined threshold and corresponds to a peak position. In addition,a mask fabricated in this way can improve the imaging performance ascompared with that fabricated according to the prior art. Each assistpattern is not particularly limited to the one almost similar to thedesired pattern, and may take any form as long as it facilitates themask fabrication.

When each assist pattern is almost similar to the pattern to betransferred by exposure, it preferably has a size around 75% that of acontact hole pattern to be transferred by exposure. The size meansherein not the area but the length of one side of the pattern. Forexample, when a square pattern 120 nm on a side is formed on the mask totransfer a 120-nm contact hole pattern by exposure, the length of oneside of each assist pattern need only be about 90 nm. Since the P map413 appropriately specifies positions to insert (arrange) assistpatterns, the insertion (arrangement) of the assist patterns leads to aconsiderable improvement in resolving power. It is therefore unnecessaryto fix the size of each assist pattern to 75% that of the contact holepattern to be transferred by exposure. According to the examinations ofvarious cases by the inventor of the present invention, a sufficienteffect of the insertion of assist patterns can be obtained even whenthey each have a size 50% to 85% that of the contact hole pattern to betransferred by exposure.

If the contact hole pattern has a rectangular shape, rectangular assistpatterns need only be inserted (arranged). The short-side length of eachof these assist patterns need only be 50% to 80% that of the contacthole pattern to be transferred by exposure.

If the pattern to be transferred by exposure is a line pattern, linearassist patterns need only be inserted. Since a linear pattern isresolved readily, the width of each assist pattern is preferably 35% to70% that of the line pattern to be transferred by exposure.

Eleventh Embodiment

Multiple exposure using a P map 413 will be explained in the eleventhembodiment. Multiple exposure in a broad sense is known as onemicropattern exposure method. The multiple exposure in a broad senseincludes multiple exposure in a narrow sense and a plurality of times ofexposure. In the multiple exposure in a narrow sense, latent imagepatterns are added without a development process. For example, in therepresentative double exposure, the mask pattern is divided into twotypes, that is, a dense pattern and sparse pattern, thereby performingdouble exposure. There is another double exposure in which a linepattern is divided into patterns in the longitudinal direction andhorizontal direction, and they are individually transferred by exposure,thereby forming a desired line pattern. In contrast, in the plurality oftimes of exposure, latent image patterns are added through a developmentprocess. These exposure schemes are approaches to reducing the k1factor, and will be merely referred to as “multiple exposure”hereinafter, including the multiple exposure in a narrow sense and theplurality of times of exposure.

The P map 413 includes regions having negative values, and has afunction of canceling imaging (i.e., the formation of an aerial image),as described above.

Pieces of P map calculation information are assumed to be the same as inthe fifth embodiment. FIG. 20 shows the defocus characteristic whenassist patterns that are in phase with a desired pattern to betransferred by exposure are inserted (arranged) at positions havingpositive values, and that when assist patterns that are in phase withthe desired pattern are inserted at positions having negative values onthe P map 413 calculated from the pieces of P map calculationinformation. FIG. 20 also shows the defocus characteristic when thereare no assist patterns. In FIG. 20, the abscissa indicates the defocusamount, and the ordinate indicates the diameter of the contact holepattern (CD).

Referring to FIG. 20, the defocus characteristic when assist patternsare inserted (arranged) at positions having positive values is betterthan that when there are no assist patterns on the P map 413. However,the defocus characteristic when assist patterns are inserted (arranged)at positions having negative values is poorer than that when there areno assist patterns on the P map 413. In this manner, FIG. 20 revealsthat increasing the number of assist patterns arranged around thepattern to be transferred by exposure is not always beneficial unlikethe conventional concept.

Positions having negative values on the P map 413 represent a forbiddenpitch. The positions having negative values on the P map 413 arevectors, which depend on the distance and direction. In this case, fourvectors, that is, (±225, 225) and (±225, −225) represent a forbiddenpitch. These vectors each point the direction from the origin to aregion which exhibits a coherence equal to or lower than a threshold onthe P map 413.

When pattern data 401 is divided so as to avoid a forbidden pitchassuming the vectors representing the forbidden pitch as referencevectors, final pattern data 401 free from any forbidden pitch can begenerated.

A process of generating pattern data 401 free from any forbidden pitchby a mask generation program 414 will be explained in detail withreference to FIG. 21. Pieces of P map calculation information areassumed to be input in advance via an input unit 50 by the user andstored in a storage unit 40. Also, the mask generation program 414 isassumed to be installed from a storage medium 70 connected to a mediuminterface 60 and stored in the storage unit 40 via a control unit 20.The mask generation program 414 is started up in response to a start-upcommand input from the input unit 50 by the user and executed by thecontrol unit 20.

In step S1102, the control unit 20 calculates the P map 413 based on thepieces of P map calculation information. Note that a mask function 412is set based not on the entire pattern data (target pattern) 401 but onits one element. More specifically, a mask function 412 is set byperforming a predetermined process for one element of the target pattern(e.g., multiplying it by a reduction magnification).

In step S1104, the control unit 20 specifies reference vectorsrepresenting a forbidden pitch from the P map 413 calculated in stepS1102. More specifically, reference vectors are specified by extractingvector quantities each from the origin to a region which exhibits acoherence equal to or lower than a threshold and corresponds to anegative peak on the P map 413.

In step S1106, the control unit 20 sets an initial value “1” to areference number i of pattern data 401 generated in a step to bedescribed later. Pattern data having the reference number i will bereferred to as the i-th pattern data hereinafter.

In step S1108, the control unit 20 checks whether the pattern data 401has a forbidden pitch. More specifically, the control unit 20 checkswhether, when an element of interest is selected from a plurality ofelements of the pattern data 401, and reference vectors are set assumingthe center of the selected element of interest as the start point, anelement exists near the end point of any of the reference vectors. Ifthe control unit 20 determines that an element exists near the end pointof any of the reference vectors, it determines that the pattern data 401has a forbidden pitch. If the control unit 20 determines that no elementexists near the end point of any of the reference vectors, it determinesthat the pattern data 401 has no forbidden pitch.

If the control unit 20 determines that the pattern data 401 has aforbidden pitch, it advances the process to step S1110; otherwise, itadvances the process to step S1112.

In step S1110, the control unit 20 removes the element near the endpoint of any of the reference vectors from the pattern data 401, andtemporarily stores information on the removed element in a cash memory.

In step S1112, the control unit 20 checks whether the determination instep S1108 has been performed for all the elements which are not removedin step S1110 of the plurality of elements of the pattern data 401.

If the control unit 20 determines that the determination in step S1108has been performed for all the elements, it advances the process to stepS1114; otherwise, it returns the process to step S1108.

In step S1114, the control unit 20 generates the i-th pattern data (i-thdata generation step). More specifically, if i=1, the control unit 20determines, as the i-th pattern data, pattern data obtained by removingall the elements near the end points of the reference vectors from thepattern data 401. If i≧2, the control unit 20 determines, as the i-thpattern data, pattern data obtained by removing all the elements nearthe end points of the reference vectors from the (i−1)-th pattern data.

In step S1116, the control unit 20 newly sets a value obtained byincrementing the reference number i of the pattern data by 1 as i.

In step S1118, the control unit 20 calculates the P map 413. Morespecifically, the control unit 20 calculates the P map 413 as thepreliminary step toward inserting (arranging) assist patterns in thepattern data 401. In step S1118, the mask function 412 is set based onall the elements of the i-th pattern data. In other words, the maskfunction 412 is set by performing a predetermined process for all theelements of the i-th pattern data (e.g., multiplying them by a reductionmagnification), thereby calculating the P map 413. Because the maskfunction 412 obtained in step S1102 is different from that obtained instep S1118, different P maps 413 must be calculated in steps S1102 andS1118.

In step S1120, the control unit 20 generates mask data 410 by inserting(arranging) assist patterns. More specifically, assist patterns areinserted (arranged) in regions each of which exhibits a value more thana predetermined threshold and corresponds to a peak position, based onthe P map 413 calculated in step S1118. The control unit 20 thendetermines data, which is obtained by including information on theassist patterns in the mask data 410, as new mask data 410. At thistime, the control unit 20 may display the mask data 410 on a displayunit 30 in place of the pattern data 401.

In step S1122, the control unit 20 checks whether there is an elementremoved from the pattern data 401 by referring to the cash memory.

If the control unit 20 determines that there is an element removed fromthe pattern data 401, it advances the process to step S1124; otherwise,it ends the process.

In step S1124, the control unit 20 generates pattern data 401, whichincludes the element removed in generating the i-th pattern data, as anew process target (second data generation step).

An example of the division of the pattern data 401 using the P map 413will be shown. Pieces of P map calculation information are assumed to bethe same as in the fifth embodiment. Consider a case in which patterndata 401 shown in FIG. 22A is the process target. The pattern data 401shown in FIG. 22A has three contact holes CP1 to CP3. The size(diameter) of each of the contact holes CP1 to CP3 is 120 nm.

The contact hole CP2 is spaced apart from the contact hole CP1 by −280nm in the y direction. The contact hole CP3 is spaced apart from thecontact hole CP2 by 225 nm in the x direction and by −225 nm in the ydirection.

The control unit 20 calculates the P map 413 by setting one element(i.e., one of the three contact holes CP1 to CP3) as the mask function412. In this embodiment, the contact hole CP1 is set as the maskfunction 412. The P map 413 has positive peaks at the positions (±280,0) and (0, ±280) on the mask. Also, the P map 413 has negative peaks atthe positions (±225, 225) and (±225, −225) on the mask.

The control unit 20 specifies reference vectors representing, forexample, a forbidden pitch from the P map 413. At this time, there arefour reference vectors (225, 225), (225, −225), (−225, 225), and (−225,−225).

The control unit 20 selects the contact hole (element) CP2 as theelement of interest assuming the pattern data 401 shown in FIG. 22A asthe process target. In this case, when the reference vectors are setassuming the position of the selected contact hole CP2 as the element ofinterest as the start point, the contact hole (element) CP3 exists nearthe end point of one of the reference vectors. Therefore, the contactholes CP2 and CP3 have a forbidden pitch relationship.

To cancel this state, the control unit 20 removes the contact hole(element) CP3 near the end point of one of the reference vectors fromthe pattern data 401 shown in FIG. 22A to generate the first patterndata shown in FIG. 22B. The control unit 20 also generates the secondpattern data shown in FIG. 22C from the contact hole (element) CP3. Withthis operation, the pattern data 401 shown in FIG. 22C is divided intothe first pattern data shown in FIG. 22B and the second pattern datashown in FIG. 22C. This division can generate mask data for two masksfree from any forbidden pitches.

As has been described in the fifth and ninth embodiments, the insertion(arrangement) of assist patterns based on the P map 413 improves theimaging performance of the mask. Hence, by inserting (arranging) bestsuited assist patterns in the first pattern data shown in FIG. 22B andthe second pattern data shown in FIG. 22C to generate the pattern datashown in FIGS. 22D and 22E, the imaging performance can be improved ascompared with simple double exposure.

When the mask data shown in FIGS. 22D and 22E are input to an EB drawingdevice, two masks based on them are fabricated. When double exposure isperformed using these two masks, contact holes CP1 to CP3 can be formedwith a higher accuracy than when the exposure is performed using a maskidentical to the pattern data shown in FIG. 22A.

Twelfth Embodiment

The optimization of the effective light source will be explained in thetwelfth embodiment. In the optimization of the effective light source,it need only be determined such that peaks (regions each of whichexhibits a value equal to or more than a predetermined threshold) on a Pmap 413 match the positions of elements of pattern data 401.

Pieces of P map calculation information other than effective lightsource information 402 are assumed to be the same as in the fifthembodiment. Consider the optimization of the effective light source forpattern data (target pattern) 401 having three contact holes CP11 toCP13, as shown in FIG. 23A. The three contact holes CP11 to CP13 arearranged at an interval dd=300 nm. The size of each of the three contactholes CP11 to CP13 is 120 nm.

FIG. 23B is a chart showing the initial value of the effective lightsource (effective light source information 402). In FIG. 23B, the whitecircular line indicates σ=1, and the white regions indicate lightirradiation portions. The pupil coordinate system is normalized suchthat the distance from the center of the circle to the center of eachpole (light irradiation portion) in the x direction is set to 0.45 andthat in the y direction is set to 0.45, and the diameter of each pole(light irradiation portion) is set to 0.3.

Based on the initial value of the effective light source shown in FIG.23B, a control unit 20 calculates a P map 413 shown in FIG. 23C. The Pmap 413 shown in FIG. 23C has positive peaks at the positions (0, ±300)and (±300, 0). The P map 413 shown in FIG. 23C is suited to a mask basedon the pattern data 401 shown in FIG. 23A. This is because the intervaldd between adjacent contact holes is 300 nm on the mask based on thepattern data 401 shown in FIG. 23A.

The control unit 20 calculates a new P map 413 by re-setting a maskfunction 412 (e.g., by setting the mask function 412 as the targetpattern itself). When assist patterns are inserted (arranged) in regionseach of which exhibits a value equal to or more than a predeterminedthreshold and corresponds to a peak position on the P map 413, mask data410 as shown in FIG. 23D can be obtained. The use of a mask based on themask data 410 shown in FIG. 23D allows forming contact holes CP11 toCP13 with high accuracy.

FIG. 24 is a flowchart for explaining a process of generating mask data410 by a mask generation program 414.

In step S1202, the control unit 20 sets the effective light sourceinformation 402.

In step S1204, the control unit 20 sets the mask function 412. Note thatthe mask function 412 is set based not on the entire target pattern buton its one element. The mask function 412 is set by performing apredetermined process for the one element (e.g., multiplying it by areduction magnification).

In step S1206, the control unit 20 calculates the P map 413 based on themask function 412 set in step S1204.

In step S1208, the control unit 20 matches the P map 413 and the contactholes CP11 to CP13 as a desired pattern to be transferred by exposure.

In step S1210, the control unit 20 determines whether the contact holesCP11 to CP13 as a desired pattern match peaks (regions each of whichexhibits a value equal to or more than a predetermined threshold) on theP map 413. If the control unit 20 determines that the contact holes CP11to CP13 as a desired pattern match peaks, it advances the process tostep S1212. If the control unit 20 determines that the contact holesCP11 to CP13 as a desired pattern does not match peaks on the P map 413,it returns the process to step S1202.

In step S1212, the control unit 20 changes the mask function 412.Although the mask function 412 is set by paying attention to one elementof the target pattern in step S1204, all the elements of the targetpattern are set as the mask function 412. For this purpose, the maskfunction 412 is set by performing a predetermined process for all theelements of the target pattern (e.g., multiplying them by a reductionmagnification).

In step S1214, the control unit 20 calculates the P map 413 based on themask function 412 set in step S1212.

In step S1216, the control unit 20 inserts (arranges) assist patternsbased on the P map 413 calculated in step S1214 to generate mask data410, and ends the process.

To optimize the effective light source, it is necessary to repeat (i.e.,loop) steps S1202 to S1210 shown in FIG. 24. The initial setting of theeffective light source (effective light source information 402) isimportant to quickly complete the loop of steps S1201 to S1210. A methodof easily calculating the initial setting of the effective light source,which can quickly complete the loop of steps S1201 to S1210, in a shortperiod of time will be explained below.

Light diffracted by the mask pattern forms a diffracted lightdistribution on the pupil plane of the projection optical system. Leta(f, g) be the amplitude of the diffracted light, as described above.The coordinates (f, g) on the pupil plane of the projection opticalsystem are also normalized assuming that the pupil size (pupil diameter)of the projection optical system is 1, as described above. Letcirc(f−f′, g−g′) be a function which takes 1 for positions that fallwithin a circle with a radius of 1 and a center at (f′, g′), and takes 0for other positions. Let w(f, g) be the weighting function of thediffracted light.

First, the control unit 20 calculates a multiple integral:

S _(raw)(f,g)=∫∫w(f,g)a(f,g)circ(f−f′,g−g′)df′dg′  (35)

for |f′|≦2 and |g′|≦2.

Next, the control unit 20 calculates:

S(f,g)=S _(raw)(f,g)circ(f,g)  (36)

Lastly, the control unit 20 determines S(f, g) calculated from equation(36) as the setting value of the effective light source.

For example, consider pattern data 401 in which contact hole patterns of5 rows and 5 columns are two-dimensionally arrayed at a cycle of 300 nm,as shown in FIG. 25A. In FIG. 25A, the ordinate indicates they-coordinate on the mask surface (unit: nm), and the abscissa indicatesthe x-coordinate on the mask surface (unit: nm). The 12th embodimentalso assumes a case in which an exposure apparatus uses a projectionoptical system having an NA of 0.73 (corresponding to NA information403), and exposure light having a wavelength of 248 nm (corresponding toλ information 404).

The control unit 20 calculates the function S(f, g) describing theeffective light source, based on equations (35) and (36). FIG. 25B showsthe effective light source described by the function S(f, g) calculatedby the control unit 20. In this embodiment, the weighting function w(f,g) is assumed to be a quadratic function which satisfies (0, 0)=0.1 andw(2, 2)=1. In FIG. 25B, the ordinate indicates the coherence factor σ inthe x direction, and the abscissa indicates the coherence factor σ inthe y direction.

Referring to FIG. 25B, the effective light source described by thefunction S(f, g) changes continuously. The effective light source shownin FIG. 25B is close to that shown in FIG. 23B. Hence, the effectivelight source shown in FIG. 25B is suitable as the initial value of theeffective light source information 402 (the setting value of theeffective light source) set in step S1202 in the loop of steps S1202 toS1210.

Thirteenth Embodiment

An exposure apparatus 100 which executes an exposure process using amask 130 fabricated based on mask data 410 generated in one of theabove-described embodiments will be explained in the thirteenthembodiment. Note that FIG. 26 is a schematic block diagram showing thearrangement of the exposure apparatus 100.

The exposure apparatus 100 is an immersion exposure apparatus whichtransfers the pattern of the mask 130 onto a wafer 150 by exposure usingthe step & scan scheme via a liquid LW supplied between a projectionoptical system 140 and the wafer 150. However, the exposure apparatus100 can adopt the step & repeat scheme or another exposure scheme.

As shown in FIG. 26, the exposure apparatus 100 includes a light source110, an illumination optical system 120, a mask stage 135 for mountingthe mask 130, the projection optical system 140, a wafer stage 155 formounting the wafer 150, a liquid supply/recovery unit 160, and a maincontrol system 170. The light source 110 and illumination optical system120 constitute an illumination apparatus which illuminates the mask 130on which a circuit pattern to be transferred is formed.

The light source 110 is an excimer laser such as an ArF excimer laserhaving a wavelength of about 193 nm or a KrF excimer laser having awavelength of about 248 nm. However, the type and number of lightsources 110 are not particularly limited. For example, an F₂ laserhaving a wavelength of about 157 nm can also be used as the light source110.

The illumination optical system 120 illuminates the mask 130 with lightfrom the light source 110. In this embodiment, the illumination opticalsystem 120 includes a beam shaping optical system 121, condensingoptical system 122, polarization control unit 123, optical integrator124, and aperture stop 125. The illumination optical system 120 alsoincludes a condenser lens 126, bending mirror 127, masking blade 128,and imaging lens 129. The illumination optical system 120 can implementvarious illumination modes such as the conventional illumination andmodified illumination (e.g., quadrupole illumination and dipoleillumination) shown in FIGS. 4A and 14A.

The beam shaping optical system 121 is, for example, a beam expanderincluding a plurality of cylindrical lenses. The beam shaping opticalsystem 121 converts the horizontal to vertical ratio of the sectionalshape of collimated light from the light source 110 into a predeterminedvalue (e.g., converts the sectional shape from a rectangle to a square).In this embodiment, the beam shaping optical system 121 shapes the lightfrom the light source 110 into that having a size and an angle ofdivergence required to illuminate the optical integrator 124.

The condensing optical system 122 includes a plurality of opticalelements, and efficiently guides the light shaped by the beam shapingoptical system 121 to the optical integrator 124. The condensing opticalsystem 122 includes, for example, a zoom lens system, and adjusts theshape and angle of the light which enters the optical integrator 124.

The polarization control unit 123 includes, for example, a polarizingelement, and is set at a position nearly conjugate to a pupil plane 142of the projection optical system 140. The polarization control unit 123controls the polarization state of a predetermined region of aneffective light source formed on the pupil plane 142 of the projectionoptical system 140.

The optical integrator 124 has a function of uniforming illuminationlight which illuminates the mask 130, converting the angulardistribution of its incident light into a positional distribution, andoutputting the obtained light. The optical integrator 124 is, forexample, a fly-eye lens having its incident surface and exit surfacewhich hold a Fourier transform relationship. The fly-eye lens is formedby combining a plurality of rod lenses (i.e., microlens elements).However, the optical integrator 124 is not particularly limited to afly-eye lens, and may be, for example, a cylindrical lens array plate inwhich sets of optical rods and diffraction gratings are arrayed to beorthogonal to each other.

The aperture stop 125 is set at a position which is immediately afterthe exit surface of the optical integrator 124 and is nearly conjugateto an effective light source formed on the pupil plane 142 of theprojection optical system 140. The aperture shape of the aperture stop125 corresponds to a light intensity distribution (i.e., an effectivelight source) formed on the pupil plane 142 of the projection opticalsystem 140. In other words, the aperture stop 125 controls the effectivelight source. The aperture stop 125 can be switched in accordance withthe illumination modes. Without an aperture stop, an effective lightsource may be formed by setting a diffractive optical element (e.g., aCGH (Computer Generated Hologram)) and a prism (e.g., a conical prism)at the preceding stage of the optical integrator 124.

The condenser lens 126 converges the light beam which has emerged from asecondary light source formed near the exit surface of the opticalintegrator 124 and passed through the aperture stop 125, and uniformlyilluminates the masking blade 128 with it via the bending mirror 127.

The masking blade 128 is set at a position nearly conjugate to the mask130, and is formed from a plurality of movable light-shielding plates.The masking blade 128 forms a nearly rectangular opening correspondingto the effective area of the projection optical system 140. The lightbeam which has passed through the masking blade 128 is used asillumination light which illuminates the mask 130.

The imaging lens 129 forms, on the mask 130, an image of the light beamwhich has passed through the opening of the masking blade 128.

The mask 130 is fabricated by a mask fabrication device such as an EBdrawing device based on mask data generated by the processing apparatus1 (mask generation program) described above, and has a circuit patternto be transferred and assist patterns. The pattern of the mask 130 mayinclude a pattern other than the mask pattern generated by theabove-described mask generation program. The mask 130 is supported anddriven by the mask stage 135. Diffracted light generated by the mask 130is projected onto the wafer 150 via the projection optical system 140.The mask 130 and wafer 150 are set to have an optically conjugaterelationship. Since the exposure apparatus 100 is of the step & scanscheme, it transfers the circuit pattern to be transferred of the mask130 onto the wafer 150 by synchronously scanning them. When the exposureapparatus 100 is of the step & repeat scheme, it performs exposure whilethe mask 130 and wafer 150 stay still.

The mask stage 135 supports the mask 130 via a mask chuck, and isconnected to a driving mechanism (not shown). The driving mechanism (notshown) is formed from, for example, a linear motor, and drives the maskstage 135 in the X-, Y-, and Z-axis directions and the rotationdirections about the respective axes. Note that the scanning directionof the mask 130 or wafer 150 on its surface is defined as the Y-axisdirection, a direction perpendicular to it is defined as the X-axisdirection, and a direction perpendicular to the surface of the mask 130or wafer 150 is defined as the Z-axis direction.

The projection optical system 140 projects the circuit pattern of themask 130 onto the wafer 150. The projection optical system 140 can be adioptric system, catadioptric system, or catoptric system. The finallens (final surface) of the projection optical system 140 is coated witha coating for reducing the influence on it exerted by the liquid LWsupplied from the liquid supply/recovery unit 160 (for protection).

The wafer 150 is a substrate onto which the circuit pattern of the mask130 is projected (transferred). However, the wafer 150 can besubstituted by a glass plate or another substrate. The wafer 150 iscoated with a resist.

The wafer stage 155 supports the wafer 150 and moves it in the X-, Y-,and Z-axis directions and the rotation directions about the respectiveaxes using a linear motor, like the mask stage 135.

The liquid supply/recovery unit 160 has a function of supplying theliquid LW to the space between the wafer 150 and the final lens (finalsurface) of the projection optical system 140. The liquidsupply/recovery unit 160 also has a function of recovering the liquid LWsupplied to the space between the wafer 150 and the final lens of theprojection optical system 140. A substance which has a hightransmittance with respect to the exposure light, prevents dirt fromadhering on the projection optical system 140 (on its final lens), andmatches the resist process is selected as the liquid LW.

The main control system 170 includes a CPU and memory and controls theoperation of the exposure apparatus 100. For example, the main controlsystem 170 is electrically connected to the mask stage 135, wafer stage155, and liquid supply/recovery unit 160, and controls the synchronousscanning between the mask stage 135 and the wafer stage 155. The maincontrol system 170 also controls the switching among the supply,recovery, and supply/recovery stop of the liquid LW, based on, forexample, the scanning direction and velocity of the wafer stage 155 inexposure. The main control system 170 receives effective light sourceinformation in one of the above-described embodiments, and controls theaperture stop, diffractive optical element, and prism to form aneffective light source. The effective light source information may beinput to the main control system 170 by the user or by transmitting theeffective light source information from the processing apparatus 1 tothe exposure apparatus 100 by connecting the processing apparatus 1 andexposure apparatus 100 to allow data communication between them. If theprocessing apparatus 1 and exposure apparatus 100 are connected to allowdata communication between them, the exposure apparatus 100 includes aknown data reception unit and the processing apparatus 1 includes aknown data transmission unit.

Although the processing apparatus 1 described above can be a computerarranged outside the exposure apparatus 100, the main control system 170can have the function of the processing apparatus 1 described above,instead. In this case, the main control system 170 can calculate a lightintensity distribution (aerial image), which is formed on the wafersurface, using a P operator in a shorter period of time. In other words,the main control system 170 can improve the speed of partial coherentimaging calculation, thus shortening the time taken for the model-basedRET. Hence, the exposure apparatus 100 can optimize the exposureconditions (e.g., optimize the effective light source for the mask 130)in a short period of time, thus improving the throughput. The maincontrol system 170 also can generate mask data which exhibits an imagingperformance more excellent than in the prior art using a P map.

In exposure, a light beam emitted by the light source 110 illuminatesthe mask 130 by the illumination optical system 120. The light beamwhich reflects the circuit pattern of the mask 130 upon beingtransmitted through it forms an image on the wafer 150 via the liquid LWby the projection optical system 140. The exposure apparatus 100 has anexcellent imaging performance and can provide devices (e.g., asemiconductor device, an LCD device, an image sensing device (e.g., aCCD), and a thin-film magnetic head) with high throughput and a goodeconomical efficiency. These devices are fabricated by a step ofexposing a substrate (e.g., a wafer or glass plate) coated with a resist(photosensitive agent) using the exposure apparatus 100, a step ofdeveloping the exposed substrate, and other known steps.

While the present invention has been described with reference toexemplary embodiments, it is to be understood that the invention is notlimited to the disclosed exemplary embodiments. The scope of thefollowing claims is to be accorded the broadest interpretation so as toencompass all such modifications and equivalent structures andfunctions.

What is claimed is:
 1. A generation method of generating, by a computer,data of a pattern of a mask used for an exposure apparatus including aprojection optical system, comprising the following steps being executedby a computer: a division step of dividing an effective light sourceformed on a pupil plane of the projection optical system into aplurality of point sources; a pupil function generation step ofgenerating a plurality of shifted pupil functions by shifting a pupilfunction corresponding to each of the plurality of point sources by ashift amount in accordance with a position of each point source, thepupil function being a function describing a pupil of the projectionoptical system; a defining step of defining a matrix by arranging eachof the plurality of shifted pupil functions generated in the pupilfunction generation step in each row or each column of the matrix; afirst calculation step of performing singular value decomposition of thematrix defined in the defining step, thereby calculating an eigenvalueand an eigenfunction; a second calculation step of calculating a maprepresenting, when elements of a target pattern are inserted on anobject plane of the projection optical system, an influence the elementsinflict on each other, based on a distribution of the light diffractedby the target pattern, and the eigenvalue and the eigenfunctioncalculated in the first calculation step; and a data generation step ofgenerating data of the pattern of the mask based on the map calculatedin the second calculation step.
 2. The method according to claim 1,wherein in the pupil function generation step, the pupil function isshifted by the shift amount equal to a difference between the centralposition of the pupil of the projection optical system and the positionof each point source.
 3. The method according to claim 1, wherein theplurality of shifted pupil functions includes information representingan aberration of the projection optical system.
 4. The method accordingto claim 1, wherein, in the pupil function generation step, theplurality of shifted pupil functions are arranged one-dimensionally. 5.The method according to claim 1, wherein the plurality of shifted pupilfunctions includes information representing at least one of apolarization state of the light which illuminates the mask, a variationin light intensity of an effective light source of the illuminationoptical system, and a diffraction efficiency of the mask.
 6. The methodaccording to claim 1, wherein in the data generation step, when regionsdetermined based on a threshold of the map do not match positions of aplurality of elements of the target pattern assuming data of the targetpattern as a process target, the effective light source is newly set,and data of the pattern of the mask is generated based on the newly seteffective light source.
 7. The method according to claim 1, wherein thedata generation step includes: a specifying step of specifying areference vector from an origin to a region which exhibits a value notless than a threshold on the map calculated in the second calculationstep; a first data generation step of selecting one of a plurality ofelements of the target pattern, and removing, when the reference vectoris set assuming the center of the element as a start point, an elementat a position which matches an end point of the reference vector,thereby generating first pattern data; and a second data generation stepof generating second pattern data formed from the element removed in thefirst data generation step.
 8. The method according to claim 7, whereinin the data generation step, the first data generation step and thesecond data generation step are repeated.
 9. A non-transitory tangiblecomputer-readable medium storing a program for making a computer executea process of generating data of a pattern of a mask used for an exposureapparatus including a projection optical system, the program making thecomputer execute: a division step of dividing an effective light sourceformed on a pupil plane of the projection optical system into aplurality of point sources; a generation step of generating a pluralityof shifted pupil functions by shifting a pupil function corresponding toeach of the plurality of point sources by a shift amount in accordancewith a position of each point source, the pupil function being afunction describing a pupil of the projection optical system; a definingstep of defining a matrix by arranging each of the plurality of shiftedpupil functions generated in the generation step in each row or eachcolumn of the matrix; a first calculation step of performing singularvalue decomposition of the matrix defined in the defining step, therebycalculating an eigenvalue and an eigenfunction; a second calculationstep of calculating a map representing, when elements of a targetpattern are inserted on an object plane of the projection opticalsystem, an influence the elements inflict on each other, based on adistribution of the light diffracted by the target pattern, and theeigenvalue and the eigenfunction calculated in the first calculationstep; and a data generation step of generating data of the pattern ofthe mask based on the map calculated in the second calculation step. 10.A mask fabrication method comprising: generating data of a pattern of amask using a generation method defined in claim 1; and fabricating amask using generated data of the pattern of the mask.
 11. An exposuremethod comprising steps of: illuminating a mask fabricated by a maskfabrication method defined in claim 10; and projecting an image of apattern of the mask onto a substrate via a projection optical system.12. A device manufacturing method comprising steps of: exposing asubstrate by an exposure method defined in claim 11; and processing theexposed substrate.